Brightness contrast–contrast induction model predictsassimilation and inverted assimilation effects

Department of Biomedical Engineering,Faculty of Engineering, Tel-Aviv University,

IsraelYuval Barkan

Department of Biomedical Engineering,Faculty of Engineering, Tel-Aviv University,

IsraelHedva Spitzer

Department of Biomedical Engineering,Faculty of Engineering, Tel-Aviv University,

IsraelShmuel Einav

In classical assimilation effects, intermediate luminance patches appear lighter when their immediate surround is comprisedof white patches and appear darker when their immediate surround is comprised of dark patches. With patches eitherdarker or lighter than both inducing patches, the direction of the brightness effect is reversed and termed as “invertedassimilation effect.” Several explanations and models have been suggested, some are relevant to specific stimulusgeometry, anchoring theory, and models that involve high level cortical processing (such as scission, etc.). None of thesestudies predicted the various types of assimilation effects and their inverted effects. We suggest here a compoundbrightness model, which is based on contrast–contrast induction (second-order adaptation mechanism). The suggestedmodel predicts the various types of brightness assimilation effects and their inverted effects. The model is composed ofthree main stages: (1) composing post-retinal second-order opponent receptive fields, (2) calculations of local and remotecontrast, and (3) adaptation of the second-order (contrast–contrast induction). We also utilize a variation of the Jacobiiteration process to enable elegant edge integration in order to evaluate the model is performance.

Keywords: computational model, assimilation, inverted assimilation, brightness, contrast–contrast induction,contrast gain control

Citation: Barkan, Y., Spitzer, H., & Einav, S. (2008). Brightness contrast–contrast induction model predicts assimilation andinverted assimilation effects. Journal of Vision, 8(7):27, 1–26, http://journalofvision.org/8/7/27/, doi:10.1167/8.7.27.

Introduction

The effects of brightness (or chromatic) assimilationand contrast induction both depend on the contextualinducing of the light or surfaces (Anstis, 2005; Hong &Shevell, 2004). Brightness induction and assimilationeffects have been regarded in the literature as two conflictingbrightness effects. The brightness induction effect (first-order induction) exhibits an appearance of a shift inbrightness away from the brightness of the surroundingsurface, whereas the assimilation effect exhibits a perceivedshift toward the brightness of adjacent surround surfaces(Hong & Shevell, 2004; Shapley & Reid, 1985). Anadditional induction effect that has been described is thecontrast–contrast induction which is the effect of modulatingthe perceived contrast of a central area due to the contrast ofits surrounding area (Olzak & Laurinen, 1999, 2005; Singer& D’Zmura, 1999; Xing & Heeger, 2001).Helson (1963) suggested that the assimilation and

contrast induction are actually continuum effects, which

probably derive from the same mechanism. He suggestedthat spatial frequency of the lines determine the type ofeffect. Due to this approach a recent computationalmodel has been suggested (Vanrell, Baldrich, Salvatella,Benavente, & Tous, 2004).One of the prominent and most investigated assimila-

tion effects is the White’s effect in lightness (White, 1979;Figure 1). In this effect the gray stripes superimposed onblack stripes appear much brighter than when they arepositioned between the black stripes. Different explan-ations have been suggested to explain this effect. Amongthe more recent explanations are

1. Anderson’s (1997) suggestion that the mechanism ofvisual “scission” the gray lines as separatetransparent layers;

2. some explanations and models of the effect relatedto the stimulus geometry (Ross & Pessoa, 2000;Todorovic, 1997; Zaidi, Spehar, & Shy, 1997).Todorovic (1997) explained, for example, theWhite’s effect using the T-junction rule;

Journal of Vision (2008) 8(7):27, 1–26 http://journalofvision.org/8/7/27/ 1

doi: 10 .1167 /8 .7 .27 ISSN 1534-7362 * ARVO

treats

Received October 13, 2007; published October 17, 2008

3. additional geometric explanations provided as acomputational model that suggest oriented elongatedreceptive fields, oriented difference of Gaussians(ODOG) (Blakeslee & McCourt, 1999; Robinson,Hammon, & de Sa, 2007); and

4. usage of filling in models (Grossberg, 2003; Ross &Pessoa, 2000).

It has been shown that in different spatial configurationsof White’s effect both brightness contrast and assimilationmechanism plays a role (Blakeslee & McCourt, 2004).However, in the classical spatial arrangement presented inFigure 1, it is likely that assimilation mechanism plays amajor role (Blakeslee & McCourt, 2004; White, 1981).It has been shown that the White’s effect occurs only

when luminance of the test regions lies between theminimum and maximum luminance regions (Spehar,Gilchrist, & Arend, 1995; Spehar, Iglesias, & Clifford,2005). An opposite effect occurs, “inverted White’seffect,” when the luminance of the test regions lies outsidethe luminance range of the surrounding strips (Figure 11A;Ripamonti & Gerbino, 2001; Spehar, Clifford, & Agostini,2002). Many of the explanations (including 2 and 4,above) and models such as the oriented filter of Blakesleeand McCourt (1999), which have been given for theclassical White’s effect, cannot account for the invertedWhite’s effect (Figure 11A). This opinion was previouslyexpressed by Spehar and colleagues (2002), who claimedthat models such as contrast-integration models whichalso account for the context (Ross & Pessoa, 2000) orT-junction (Todorovic, 1997; Zaidi et al., 1997) would beunsuccessful in including the inverted White’s effect.Furthermore, Howe (2005) presented a circular variant ofWhite’s effect in which all the junctions have beenremoved without significantly affecting the strength ofthe illusion, suggesting that the junctions are not animportant consideration in all versions of White’s effect.

Rippamonti and Gebino (2001), who were aware ofWhite’s effect and its inverted effect, suggested that theinverted White’s effect appears when the contrast betweenthe test stripes and adjacent strips is more prominent; theassimilation is more pronounced when the test stripes arecloser in their luminance to the adjacent stripes. In ourview, this observation has to be scrutinized but withadditional findings, such as that both White’s and itsinverted effect, causes a relatively stronger brightness shiftat high spatial frequency (McCourt, 2005).Branka Spehar and colleagues (2002) performed psy-

chophysical experiments to test whether the White’s effectand the inverted White’s effect have the same or separatemechanisms. They found and claimed that direction of theWhite’s effect does not depend on the amount of black orwhite borders in the immediate adjacent area to the testregion, while the direction of the effect is changed in theinverted White’s effect at different spatial conditions.These results led the authors to suggest that these twoeffects are governed by different mechanisms, and theysuggested that the White’s effect involves more spatialglobal mechanisms. We test their suggestion in our study,by predictions of both effects, on their specific White’sand inverted White’s effect variations, through the singlemechanism of our model.Research of the assimilation effect has been done on

additional types of stimuli, for example, on “Stuart’sRings” (Anstis, 2005) and achromatic rings patterns(Hong & Shevell, 2004). Explanations for these and otherappearance assimilation effects, which are not included inthe explanations of White’s effect, have also beensuggested by other research groups. Among these, it hasbeen suggested that the brightness of an object isdetermined by a combination of neuronal signals through-out the visual stimulus (Hong & Shevell, 2004; Reid &Shapley, 1988; Shevell, Holliday, & Whittle, 1992). Somestudies suggested specific explanations that are built onthe a mechanism of weighted sum of contrasts fromdifferent locations of the background (Rudd & Arrington,2001; Zaidi, Yoshimi, Flanigan, & Canova, 1992; Zaidi &Zipser, 1993). Hong and Shevell (2004) claimed that allthese previous explanations cannot account for theperceived brightness in ring stimuli (in Figure 2 of Hong& Shevell, 2004; Figure 8A). They suggested, however,that only a model which postulates also the influence of anon-contiguous edge has the potential to explain theperceived brightness obtained by the different ring stimuli(Hong & Shevell, 2004).Spehar and colleagues suggested that assimilation

effects are obtained when the target luminance isintermediate in luminance to the different inducingregions. Otherwise, contrast effects are observed (Speharet al., 1995, 2005; Spehar & Zaidi, 1997).Current computational models have not succeeded in

predicting the different types of the brightness assimilationeffects (Anstis, 2005; Hong & Shevell, 2004; White, 1979)and the “inverted assimilation” effects. The computational

Figure 1. Classical White’s effect. The right gray bars appearmuch darker than the left gray bars, even though they areidentical.

Journal of Vision (2008) 8(7):27, 1–26 Barkan, Spitzer, & Einav 2

model of Blakeslee and McCourt (2004) and the moreelaborated version of this ODOG model (Robinson et al.,2007) cannot account for all of the geometrical variationsof the effect, which appear also with random patterns(Anstis, 2005). As far as we know, there is no suggestedcomputational model in the literature which succeeded inpredicting the “conflicting” effects of assimilation,“inverted assimilation,” or even one of set of theseeffects such as the White’s effect and its inverted effect(Figures 7A and 11A). Our suggested model presents asingle mechanism and computational model that showspredictions of these different brightness effects, whichare perceived as opposite effects.

Model

The model aimed at suggesting a post-retinal mecha-nism for performing achromatic induction of the secondorder, i.e., adaptation of the second order. The model iscomposed of three main stages. The first stage describesthe transformation of the visual stimulus into the responseof post-retinal second-order opponent receptive fields(SORFs). These SORFs might refer to cortical levels eventhough the receptive fields are not necessarily oriented.These SORFs have various spatial resolutions, in com-pliance with the diversity found in the visual system. Thesecond stage of the model describes the calculations oflocal and remote contrast, based on the multi-scale SORFresponses. The third stage describes the adaptation ofthe second order (contrast–contrast induction) (Chubb,Sperling, & Solomon, 1989; Spitzer & Barkan, 2005; Xing& Heeger, 2001). This adaptation mechanism plays a roleas a contrast gain control and is expressed by the adaptedresponses of the SORF cells. In order to evaluate theperformance of the computational model, we performed atransformation of the adapted SORF cells’ responses to aperceived image in a standard intensity space.

Response of the second-order opponentreceptive field

The SORF cells receive their input from the retinalganglion cells through several processing layers. Theretinal ganglion cells perform an adaptation of the first-order and the SORF cells receive their responses after theadaptation of the first order. This model focuses on theadaptation of the second order (contrast–contrast induc-tion) due to the current scope of the paper. In the currentscope, when only brightness contrast–contrast induction isexamined, the brightness first-order adaptation is notmodeled for the sake of simplicity. Accordingly the effectof brightness simultaneous contrast is not processed heresince it is related to the adaptation of the first order(Spitzer & Barkan, 2005).

The SORF cells have an opponent type receptive fieldwith a center-surround spatial structure. Since adaptationof the first order is not modeled here, the RF input wastaken from signals of photoreceptors rather than fromretinal opponent cells responses.The spatial response profile of the two subregions of the

SORF, “center” and “surround,” is expressed by thecommonly used difference-of-Gaussians (DOG).The “center” signals that feed the SORF are defined as

an integral of the photoreceptors quantum catches over thecenter subregion, with a Gaussian decaying spatial weightfunction, fc(x j x0, y j y0) (Shapley & Enroth-Cugell,1984):

Lkcenðx0; y0Þ ¼ZZ

cen<area

Lphoto�rðx; yÞ I f kc ðxj x0; yj y0Þ I dx I dy;

ð1Þ

while the Lphoto-r at locations x0, y0 represents the responseof the center area of the SORF, which is centered atlocation x0, y0. The following equations are similarlyexpressed, but for the sake of simplicity, x0, y0 at thefollowing equations are substituted by zero: x0 = y0 = 0. kis an index that represents the specific spatial resolution,where k = 1 is the finest resolution. fc is defined as:

f kc x; yð Þ ¼ expðj x2 þ y2=> k2

cenÞ: I > k

x;cen

; y Z center area; ð2Þ

where > represents the radius of the center region of thereceptive field. The “Surround” signals are similarlydefined, with a spatial weight function three times largerin diameter than that of the “center.”

Lksrndðx0; y0Þ ¼ZZ

surround<area

Lphoto<rðx; yÞ I f ks ðx; yÞ I dx I dy; ð3Þ

where fs is defined as a decaying Gaussian over thesurround region:

f ks x; yð Þ ¼ expðj x2 þ y2=>k2

surÞ: I >ksur

; x; y Z surround area: ð4Þ

The total weight of fc and fs is 1.The response of the SORF, Lsorf, is the subtraction of

the “center” and “surround” signals:

Lksorf ¼ Lkcen j Lksrnd: ð5Þ

The calculations of the SORF response might bealternatively derived from an achromatic double-opponent

Journal of Vision (2008) 8(7):27, 1–26 Barkan, Spitzer, & Einav 3

RF structure, where the center and the surround regionscontained opponent subunits as has been suggested fromchromatic double-opponent RFs (Spitzer & Barkan, 2005).

Local and remote contrast

We propose that the adaptation mechanism acts as aremote contrast gain control on the SORF response. It istherefore necessary to calculate local and remote con-trasts. We suggest that contrast calculation can beobtained from the multi-scale SORF.Since the presented model processes contrast domain, it

is reasonable to suggest a gain control mechanism that isdependent on zones which are spatially larger than edgeswhich are commonly modeled as classical opponentreceptive fields. A single classical opponent RF cannotaccount as a measure to contrast, as it only measures alocal edge of a stimulus. We propose, therefore, that localand remote contrasts are measured within local andremote regions. In these regions the contrast is measuredby considering a variability of opponent RF sizes whichrepresent opponent RF with different scales.Our model presumes that in the remote area there is

higher sensitivity to higher contrasts than to low contrastin the same spatial location. Therefore, the contrastintegration across the remote area of the opponent RF isnot necessarily a linear spatial integration. Such a non-linear integration, as suggested here, in where the higherresponses to higher contrast obtain larger weight-functionin the spatial integration operation, has physiologicalplausibility (Discussion).

“Local” contrast

The suggested local contrast calculation is performed bycalculating a local smoothing absolute response for eachresolution and by summing these responses, Equation 6.Use of absolute signals enables the system to relate to theachromatic SORF responses as a contrast quantifier(Spitzer & Barkan, 2005; Spitzer & Hochstein, 1985).

Clocal x; yð Þ ¼X

k¼all scales

Z Zlocal<area

��Lksorfðx; yÞ�� IWkðx; yÞ I dx I dyZ Z

local<area

Wkðx; yÞ I dx I dy;

ð6Þ

where Wk is a non-linear spatial weight function which isdependent also on the local opponent response, Lsorf

k .(Equation 7). Due to the nature of opponent signals(Equation 5) which perform edge detection at differentresolutions, they yield a signal only at locations adjacentto edges. Thus, a common local averaging of such signalswould lead to an erroneous low value of average contrast

as it would always consider low values that derive fromlocations that are not adjacent to edges. In order toovercome that obstacle, we propose that in this local area,higher SORF response obtain higher weight function,when calculating their contribution to the local contrast.

Wk x; yð Þ ¼ ��Lksorfðx; yÞ�� I exp jx2 þ y2

>2local

� �: ð7Þ

“Remote” contrast

The remote contrast represents the peripheral area thatextends beyond the borders of the opponent classicalreceptive field (Creutzfeldt, Crook, Kastner, Li, & Pei,1991; Creutzfeldt, Kastner, Pei, &Valberg, 1991; Solomon,Peirce, & Lennie, 2004). The “remote” area is composedof an annulus-like shape around the entire RF region(Spitzer & Barkan, 2005). The second-order remote signal(Cremote) represents the average contrast in this remote area.The remote signal is defined as an inner product of the localcontrast (Clocal, Equation 6) response at each location of theremote area with a non-linear decaying Gaussian weightfunction (Wc) (or as a convolution for all spatial locations).This weight function obtains higher values for higher localcontrast values (as was also explained in local contrastcalculations, Equation 6).

Cremote x; yð Þ ¼

Z Zremote<area

��Clocalðx; yÞ�� IWcðx; yÞ I dx I dy

Z Zremote<area

Wcðx; yÞ I dx I dy;

ð8Þ

Wc x; yð Þ ¼ Clocal x; yð Þ I exp jx2 þ y2

>2remote

� �: ð9Þ

Adaptation of the second order

We propose that the result of the second-order adapta-tion can be expressed as a simple gain control mechanism.The local gain factor is determined by the ratio of localcontrast and remote contrast. The gain factor (GF) istherefore defined as follows:

GF x; yð Þ ¼ Clocalðx; yÞ þ "

Cremoteðx; yÞ þ "; ð10Þ

where " is a constant that determines the degree of gain.The response of each SORF cell after adaptation istherefor:

Lksorf<adaptedðx; yÞ ¼ GFðx; yÞ I Lksorf : ð11Þ

Journal of Vision (2008) 8(7):27, 1–26 Barkan, Spitzer, & Einav 4

These SORF responses are the result of the adaptationand are used by higher visual stages as inputs for furtherprocessing. In order to evaluate how these signals areinterpreted, a transformation from the SORF responses toan achromatic space is required.

Methods

This section describes the different parameters used inthe model simulation and also presents intermediateresults for enhancing the clarity of the model. In thissection, we demonstrate the model performance on aspecific exampleVthe well known White’s effect image(Figure 1, at a resolution of 300 � 400 pixels).Intermediate model calculations and parameters will beintroduced throughout this example. The same parameterswere used for all simulated images, which are presented inthe Results section.The model parameters were chosen to be roughly in the

range of physiological findings, at least for parameterswhich have been studied.

Response of the second-order opponentreceptive field

The “center” signal represents the response of the centralarea of each SORF. At the finest resolution (k = 1), thiscentral area is simulated by a single pixel. Accordingly,the “center” size for the finest grid was chosen as 1 pixel,i.e., Lcen (Equation 1) is equal to the original imageluminance (at each spatial location). For different kvalues, the center size was chosen as a diameter of kpixels. The “surround” diameter (Equations 3 and 4) waschosen as 3 times larger than the center diameter (Shapley& Enroth-Cugell, 1984). This ratio was applied for all kresolutions. The “center” and “surround” calculationswere applied, and Lsorf

k calculation (Equation 5) wasapplied by a convolution of the original luminance valueswith a filter that represent the subtraction of center andsurround contributions, for example, when k = 1 theconvolution kernel is:

j 0:1095 j 0:1405 j 0:1095j 0:1405 1 j 0:1405j 0:1095 j 0:1405 j 0:1095

0@

1A: ð12Þ

The SORF response (before adaptation; Equation 5) wastested on the White’s effect image, and the result for k = 1is presented in Figure 2. As expected, the responses havebeen obtained only adjacently to the edges, where thebackground stripes (black and white in the originalimage; Figure 1) yield higher “gradient” response (blueand red striped in Figure 2) than the responses to the

stripes along the location of the “test” stripes (cyan andyellow in Figure 2).

Local and remote contrast

The next step in the model is the calculation of the localand remote contrast, in order to process the texture stimuli(contrast).

Calculation of local contrast

The calculation of Clocal (Equation 6) was performedwith a >local diameter of 10 pixels for all resolutions. Inorder to enable the influence of more distant edges, thecutoff of this calculation was taken in a diameter of5* >local. This large spatial range of cutoff is required dueto the significant weight given to high contrast values evenat distant locations (Equation 7). The resolutions wherecalculated with k = 1, 3, 5, 7 (model, Equations 1–5).The intermediate result of the local contrast (Clocal) of

the White’s effect is demonstrated in Figure 3. The figureclearly demonstrates the different local contrast regions.The white background represents the high backgroundcontrast, while the two gray squares represent the lowercontrast squares, derived as a result of regions with thegray stripes. Note that the squares’ contrast differ fromeach other due to the value of the gray stripe which is notthe median value between the values of the white and theblack stripes.

Figure 2. Absolute SORF response. Illustration of an intermediatestage of the model, of the absolute SORF response, Lsorf(Equation 5), to the White’s effect image at each x, y location.The color at each location presents the response magnitude ofeach opponent receptive field according to the color bar on theright side of the figure. The response magnitude in the “test”region is smaller (yellow and cyan lines) than the responsemagnitude at the background (blue and red lines) (see detailedexplanation in the text).

Journal of Vision (2008) 8(7):27, 1–26 Barkan, Spitzer, & Einav 5

Calculation of remote contrast

The calculation of Cremote (Equation 8) was performedwith a >remote diameter of 10 pixels for all resolutions. Thecutoff of this calculation was taken as a diameter 15*>remote. This chosen “remote” size is also within the range ofthe reported electrophysiological findings (Creutzfeldt,Crook, et al., 1991; Creutzfeldt, Kastner, et al., 1991). Theintermediate result of this remote contrast is demonstratedin Figure 4. It can be shown that the boarders of the squaresare not existent any more due to the larger spatial weightfunction. Note that remote contrast is calculated over thelocal contrast response (Equation 6).The Cremote signals have a smaller response range than

the range of local signals (Figure 4) due to the largeintegration area of the Cremote signal. Since the Cremote

signal calculation in the “test” region includes also thebackground contrast (the white area in Figure 4), the resultof the Cremote signal is an average of the contrasts of both“test” and “background” regions. Consequently, theCremote signal in the “test” area obtains higher values thanthe local contrast at the same “test” area. In other cases(such as in the inverted White’s effect; Figure 11), theCremote signal of the “test” area can obtain lower valuesthan the local contrast signal at the same “test” area.

Adapted SORF response

The adapted SORF signals, Lsorf-adapted, are then calcu-lated (Equations 10–11) with " = 0.4. A smaller value ofbeta causes a stronger contrast–contrast induction.

Both the adapted SORF absolute responses and theoriginal SORF absolute responses are plotted, while k = 1,along a vertical line located at x = 300 (yellow vertical linein Figure 7A) in Figure 5. The adapted SORF, Lsorf-adapted,presents the perceived local DOG, Equation 11. There-fore, suppression or facilitation of these responses, in

Figure 3. Local contrast. Illustration of an intermediate stage of themodel, local contrast (Equation 6) which has been performed onthe White’s effect image at each x, y location. The contrast degreeis presented by the luminance value (color bar on the right) ateach location. The dark squares present the texture of the regionswith gray stripes. Note that the darker square has been obtaineddue to smaller contrast of the gray stripes while they are adjacentto the white stripes.

Figure 4. Remote contrast. Illustration of an additional intermedi-ate stage of the model, remote contrast. The “remote” contrast(Equation 8) is higher in the background than in the “test” regions.Note that the border of the test regions is blurred due to the largerarea the remote signals extends. The remote contrast ispresented in a similar format as presented in Figure 3.

Figure 5. Adapted SORF response. Illustration of the final stageof the model (Equation 11) performed on the White’s effectimage. The figure presents the suppression of the perceivedcontrast in the test area due to the adaptation of the second order.Absolute values of SORF responses (k = 1), Lsorf (blue line,Equation 5), and calculated absolute perceived opponent response,Lsorf-adapted (red dotted line, Equation 11) are plotted along locationx = 300 (vertical yellow line in Figure 7A). The perceived contrast,Lsorf-adapted (red dotted line), in the region of the test area(y = 110–240), is lower than the original “contrast,” Lsorf (blue line).

Journal of Vision (2008) 8(7):27, 1–26 Barkan, Spitzer, & Einav 6

comparison to the original DOG (Lop), presents contrastsuppression (assimilation effects) or contrast facilitation(inverted assimilation effects) respectively. Figure 5 dem-onstrates the contrast suppression in the “test” area,which predicts the assimilation effect. In order to trans-form these perceived values, Lsorf-adapted, into perceivedluminance values, Lper, a further inverse function has to beperformed.

Transformation of the adapted color cells’response to a perceived image (inversefunction)

In order to evaluate the performance of our model, wechose to perform a transformation from the SORF valuesto intensity values (Lper) to reflect the perceived lumi-nance values. These values are presented as perceivedimages (Figures 7–17, results). Since the gain factor(Equation 10) is uniform over all resolutions, it isplausible to demonstrate the inverse function only throughthe finest resolution (k = 1). Since this stage is presentedmainly for evaluation of the model, different numericalapproaches could be applied at this stage.For the sake of simplicity and clarity of the calculations

the finest resolutionwas taken as a discrete Laplace operator,which is commonly used as an approximation to DOGfunction. The finest resolution was, therefore, calculated as aconvolution of the original image with a kernel of:

0 j 1=4 0

j 1=4 1 j 1=40 j 1=4 0

0@

1A: ð13Þ

The adapted response (Lsorf-adapted) is approximately aconvolution of the above kernel (Equation 13) with Lper,therefore,

Lk¼1ðsorf�adaptedÞðn;mÞ , Lper ðn;mÞj 1=4 I ðLperðnj 1;mÞ

þ Lperðn;mj 1Þ þ Lperðnþ 1;mÞ þ Lperðn;mþ 1ÞÞ;ð14Þ

where n, m represent the location index. An extraction ofLper would yield an iterative function:

Liþ1per ðn;mÞ ¼ Lk¼1

sorf�adaptedðn;mÞ þ 1=4 I ðLiper ðnj 1;mÞ

þ Liperðn;mj 1Þ þ Liperðnþ 1;mÞ þ Liperðn;mþ 1ÞÞ;ð15Þ

where i represents the ith iteration. This is in fact, avariation of the Jacobi method (Quarteroni & Valli, 1994).

The first guessed perceived luminance, Lper(0) , at the

inverse function according to the variation of the Jacobimethod was chosen as the original image at all locations,for this example. After applying Equation 15, theestimated perceived luminance iterations: Lper

(1) , Lper(100),

and Lper(1000) are obtained and shown in Figure 6. After

100 iterations (Figure 6B), the process starts to converge.

Figure 6. Calculation of perceived luminance. The figure presentsiterations 1,100, and 1000 (A, B, and C, correspondingly). A isthe original image. The convergence of the iterations can beseen at C.

Journal of Vision (2008) 8(7):27, 1–26 Barkan, Spitzer, & Einav 7

Iteration 1000 (Figure 6C) shows the predicted luminanceimage. At this stage, we emphasize the feasibility of themodel and not the efficiency of this inverse procedure.

Results

Our model predictions are presented for a variety ofbrightness assimilation effects and contrast–contrastinduction effects. The results are presented first, for the

well known White’s effect (Figure 7; White, 1979), therecent Hong and Shevell’s achromatic ring patterns(Figure 8; Hong & Shevell, 2004), the random Stuart’srings (Figure 9; Anstis, 2005), and a new “liquefied” imagewhich demonstrates the assimilation effect (Figure 10). Weadded the “liquefied” image and its inverted assimilationimage (not presented) in order to demonstrate the general-ization of the effects that are not connected to specificperiodicity, oriented lines or expected shapes. Theprediction of the “inverted White’s effect,” that was firstdescribed by Rippamonti and Gebino (2001), is alsoshown (Figure 11). In addition, we present the model

Figure 7. White’s effect, and its model predication. Left column (A) shows the original achromatic image. The central column (B) presentsthe image predicted by the model. The right column (C) presents the normalized luminance of the original luminance (blue line) andpredicted perceived luminance (red line) along green axes in A (upper graph) and along the yellow axes in A (lower graph). The predictedperceived luminance (red line in C) is lower than the original luminance (blue line in C) in the test stripes along the yellow axes and viceversa for the perceived luminance along the green axes.

Journal of Vision (2008) 8(7):27, 1–26 Barkan, Spitzer, & Einav 8

prediction of new inverted effects, that we have found, the“inverted achromatic ring patterns” (Figure 12) and“inverted Stuart rings” (Figure 13).To further challenge the model, we present predictions

to a challenging stimuli presented by Bindman and Chubb(2004), the bull’s-eye illusion and checkerboard contrastillusion (DeValois & DeValois, 1988; Gilchrist et al.,1999; Figures 14 and 15, respectively). In addition, wepresent the model predictions to controlled stimulisuggested by Spehar and colleagues (Spehar, Debonet, &Zaidi, 1996; Spehar & Zaidi, 1997; Figures 16 and 17).All of the predictions were simulated with the same set ofmodel parameters (Methods).The results show that the model succeeded in predicting

all of these effects correctly, with the same set ofparameters (Figures 7–17). The images and the curveresults of all the tested assimilation effects show the

correct direction of the brightness shift. Figures 7–10show the prediction of higher luminance of the graysurfaces, while their immediate surround is white and viceversa. This trend of result has also been obtained for thenew “liquefied” image (Figure 10), checkerboard contrastillusion (Figure 14), and bull’s-eye illusion (Figure 15).The same trend has also been obtained for the controlledstimuli of Spehar and colleagues (Figures 16 and 17).In all of the tested inverted assimilation effects

(Figures 11–13), the model images results and curvesshow the correct prediction of the opposite effect, i.e., thepredicted test surfaces (highest luminance) obtainedhigher values, while the immediate surround region isblack rather than when their immediate surround is gray.In all of the model predictions of the assimilation

effects (Figures 7–10) and inverted assimilation effects(Figures 11–13), we can observe that the predicted

Figure 8. Shevell’s achromatic ring patterns rings and its model predication. See description in Figure 7.

Journal of Vision (2008) 8(7):27, 1–26 Barkan, Spitzer, & Einav 9

perceived brightness shift is larger in the “test” regions(gray regions) than in the background (black and white)regions. This trend of results is demonstrated in the redcurves in comparison to the blue curves in Figures 7C–13C.

Discussion

We developed a brightness adaptation model of thesecond order for the different brightness assimilation effectsand their inverted effects. The model is based on physio-logical building blocks, second-order receptive fields(SORF), and their adaptation mechanisms of gain control.The model succeeds in predicting different assimilation

effects and inverted assimilation effect with a singlegain control mechanism of contrast–contrast induction.

However, the presented model cannot predict simulta-neous contrast effects.The model predictions support the idea that a single

second-order induction mechanism (contrast–contrastinduction) rules opposite perceived effects, i.e., assimila-tion and “inverted” assimilation effects. Although simulta-neous contrast effects express similarity to the “inverted”assimilation effect regarding the direction of the brightnessshift, we suggest that these two effects are distinguished andderive from two different mechanisms. The simultaneouscontrast effect is derived from the first-order adaptation andcan be seen as induction of the first order, while the“inverted” assimilation effect is derived from the adapta-tion of the second order (induction of the second order).In order to evaluate the model performance in image

presentation (luminance domain) rather than only SORFresponse, we propose a numeric inverse function. The

Figure 9. “Stuart rings” and its model predication. See Figure 7.

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inverse function includes a variation of Jacobi iterationprocess (Quarteroni & Valli, 1994), which enables aninverse of the model outputs after adaptation to theperceived image, including the transformation of “gradient”values into luminance values. This suggested numericmethod is one of many possible methods. We choose thismethod due to its relatively simple implementation.Our model contains several physiologically based

elements and building blocks, and some plausible physio-logical mechanisms. The post-retinal receptive fieldsmodeled here have been suggested as having balancedcenter-surround regions.The post-retinal receptive fields building blocks can also

be built from basic components of non-balanced centerand surround regions, which are more physiologicallyplausible, as found for the retinal receptive fields (Croner& Kaplan, 1995). We have also tested implementing the

unbalanced opponent response as the input to balancedsecond-order receptive fields and found a similar trend ofresults. This type of second-order receptive field wouldresemble the chromatic double-opponent structure thatwas suggested also previously for the chromatic second-order induction (Spitzer & Barkan, 2005). We chose thefirst method because it leads to relatively simple calcu-lations, and more crucial, it enables faster and smootheriterative inverse function.In addition, the suggested model consideration of

having higher sensitivity to higher remote contrasts thanto low contrast in the same spatial location has to besystematically tested psychophysically. Such an effect canbe derived from electrophysiological mechanisms, such asa mechanism of change in a conductivity of excitatory ionchannels, which depend on the electrical potential (Hodgkin-Huxley model).

Figure 10. Modified White’s effect and its model predication. See Figure 7.

Journal of Vision (2008) 8(7):27, 1–26 Barkan, Spitzer, & Einav 11

We tested the model predictions for a variety ofbrightness assimilation effects, among them: White’seffect (White, 1979; Figure 7), achromatic ring patterns(Hong & Shevell, 2004; Figure 8), random “Stuart rings”(Anstis, 2005; Figure 9), and a new display of “liquefied”patterns (Figure 10).Anstis (2005) presented a challenging stimulus and

wrote: “Some geometric theories of White’s effect invokethe role of T-junctions in the stimulus, or of elongatedreceptive fields in the stimulus. However, a new isotropicbrightness illusion called “Stuart’s Rings,” which can be

stronger than White’s effect, seems to rule out thesetheories.” Figure 9 demonstrates the performance of ourmodel on this type of random stimuli. It demonstrates thecorrect prediction of perceived luminance shift to this typeof assimilation stimulus and its inverted effect (Figure 13).The model also predicted, correctly, additional randomassimilation patterns that are designed from curved patternsFigure 10.In addition, we tested the model predictions for the above

assimilation inverted effects, such as the inverted White’seffect (Ripamonti & Gerbino, 2001; Figure 11) and new

Figure 11. Inverted White’s effect and its model predication. Left column (A) shows the original achromatic image. Note that the invertedeffect is obtained when the test stimuli luminance is outside the range of the background stripes luminance. In this case, the contrast in thetest area is stronger than the contrast in the background stripes. The central column (B) presents the image predicted by the model. Theright column (C) presents the normalized luminance of the original luminance (blue line) and predicted perceived luminance (red line)along green axes in 9A (upper graph) and along the yellow axes in 9A (lower graph). The predicted perceived luminance (red line) ishigher than the original luminance (blue line) in the test stripes along both yellow and green axes, but higher along the yellow axes (wherethe original contrast is stronger) compared to the test area along the green axes.

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inverted assimilation effects, among them inverted achro-matic ring patterns and inverted Stuart rings (Figures 11–13).The results show that the model succeeded in predicting allthese effects correctly, with the same mechanism, model,and with the same set of parameters (Figures 7–17). Thepredicted images and curves of all the tested assimilationeffects show the correct direction shift of brightness. Itshows in all cases higher luminance of the gray surfaceswhile the immediate surround are the white zones and lowerluminance while the immediate surround are the blackzones. The curves show in all the tested inverted assimilationcurves the corrected prediction of opposite effect, i.e., thepredicted white zone obtained higher values while theimmediate surround region were the black zones and havehigher values than in the case where the white zonesimmediate surround region are the gray zones.

Research of the assimilation effects in recent years posechallenges to existing and future explanations of theeffect. These studies present specific “difficult” stimulithat show the assimilation effect or its inverted effect.These stimuli could not be predicted by previouscomputational models and explanation for the assimilationeffects. Two of these stimuli are presented in the study ofBindman and Chubb (2004). They wrote in reference tothe failure of different brightness models in predicting thebull’s-eye assimilation effects (Figure 14) that pose animportant challenge for the brightness model. We testedthese types of bull’s-eye assimilation effect stimuli and thecheckerboard contrast illusion (DeValois & DeValois, 1988;Gilchrist et al., 1999; Figures 14 and 15, correspondingly)and showed the correct predictions of our model for thesestimuli.

Figure 12. “Inverted” Shevell’s achromatic ring patterns rings and its model predication. See description in Figure 11.

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Figure 18 in the discussion shows the results regardingthe classical contrast–contrast effect (induction of thesecond order, similarly as has been shown with similarchromatic stimuli (Spitzer & Barkan, 2005). These resultsof contrast–contrast induction predicted by the samemodel support the basic assumptions of this model.In the Discussion section we added three additional

stimuli, which are referred to in the literature as uniquestimuli. Figures 19–21 show stimuli of Spehar andcolleagues (Spehar et al., 1996; Spehar & Zaidi, 1997)who refer to them as “phenomena of perceived contrastand brightness.” These stimuli present an effect that isyielded from different background contrast levels oftexture but have identical luminance values.Figures 19–21 also present the successful predictions of

all tested target stimuli. The trend of the model resultsshow that the mechanism of induction of the second ordercan solely predict the correct trend of the perceived

results. Thus, we conclude that the adaptation of secondorder plays the major role in these effects (Figures 19–21).However, additional mechanism may contribute to theseeffects.This prediction is in conflict with the previous argument

that suggested that the effect in these configurations(Figures 19–21) cannot be explained in terms of simplelateral contrast–gain control (Spehar & Zaidi, 1997) asactually suggested in this model.We suggest that the above effects cannot be derived

from induction of the first order since the averageluminance is identical in all stimulus images. However,the results of the model show that contrast gain controlsuffices in order to predict this trend of effects.The rational of these predictions is that the perceived

luminance is a result of both the contrast–contrastinduction and the edge integration (part of the inversefunction, Equations 11–15). See, for example, Figure 20,

Figure 13. Inverted “Stuart rings” and its model predication. See description in Figure 11.

Journal of Vision (2008) 8(7):27, 1–26 Barkan, Spitzer, & Einav 14

presuming all four target rectangles are suppressed bysimilar contrast induction. This induction is derived fromthe remote contrast that is similar for the four stimuli targetsdue to the non-linearity integration of the remote area(Equation 8). Since the remote contrast is almost identicaland larger than the local contrast of target’s edge, thecontrast gain control will cause contrast suppression for alltarget stimuli. However, the direction of the perceivedluminance shift is determined by the luminance of thecontact area. Accordingly, the right target stimuli obtaindark shift of their appearance since the longer horizontalcontacting stripes are darker in each location than theluminance of the target stimuli. In the left target stimulithere is no significant luminance shift, since the longerhorizontal contacting stripes contain contradicting higherand lower luminance values than the gray target stimuli.The model for the different assimilation effects and their

inverted effects is based on the same suggested mecha-

nism that activates the contrast–contrast phenomenon.Two previous studies suggested the idea that a textureappears to be perceived with lower contrast whensurrounded by high-contrast texture, and same mechanismis accounted for assimilation effects (Bindman & Chubb,2004; Guclu & Farell, 2005). This contrast–contrastmechanism activates the dual effects (suppression andenhancement) of the contrast induction effects Figure 18.We suggest here that the assimilation effects and theirinverted effects can be comprehended as a specific case ofthe dual contrast–contrast effects.We would like to emphasize that we believe that the

induction of the first order (simultaneous contrast) is amechanism that enhances the relevant brightness or colorand not the texture, i.e., contrast–contrast domain. Therole of the adaptation of the second order is to enhance thetexture or contrast differences between object or surfaceand its context. It has to be note that these two processes

Figure 14. Bulls Eye and its model predication. See description in Figure 7.

Journal of Vision (2008) 8(7):27, 1–26 Barkan, Spitzer, & Einav 15

appears as contradicting each other at the achromaticdomain, while in chromatic domain they can bothcoexisted (Anstis, 2005).The contrast–contrast effects have been investigated

psychophysically thoroughly and have been summarizedby Bidman and Cubb (2004), which related only to thesuppression aspect of the effect: “medium contrast texturepatch surrounded by high contrast texture appears to havea lower contrast than an identical patch surrounded by lowcontrast texture (Cannon & Fullenkamp, 1996; Chubbet al., 1989; Olzak & Laurinen, 1999; Singer & D’Zmura,1994; Solomon, Sperling, & Chubb, 1993; Spehar et al.,1995).” The enhancement aspect of the effect has beenreported less intensively; it is the opposite effect whichshows that a test patch contrast can be enhanced bycontext lower contrast (Olzak & Laurinen, 2005; Spitzer& Barkan, 2005; Xing & Heeger, 2001).

The chosen specific procedure to perform the inversefunction has been determined due to the iteration of Jacobi(Quarteroni & Valli, 1994), which enabled us to apply a

Figure 15. Checkerboard contrast illusion and its model predication. See description in Figure 7.

Figure 16. The classical White’s effect and its inverted White’seffect in a controlled configuration (Spehar et al., 2002) and themodel predictions (right column). The specific configuration issuch that the effect is manifested on test stimuli that have thesame contrast range among the inducing stripes in both classicaland inverted variants. The results are presented along the coloredaxis in the same format as Figure 7. The model’s predictions showthe correct trend for all the test stripes. The capital letters (A1/2–D1/2) indicate the specific locations in the original stimulus imageand in the result curves, correspondingly. Note that in the originalstimulus (left column), identical capital letters (A1/2–D1/2) refer toequal intensity values.

Journal of Vision (2008) 8(7):27, 1–26 Barkan, Spitzer, & Einav 16

Journal of Vision (2008) 8(7):27, 1–26 Barkan, Spitzer, & Einav 17

Journal of Vision (2008) 8(7):27, 1–26 Barkan, Spitzer, & Einav 18

variation of a numerical algebraic mathematical tool toperform a “filling-in” process that is required in order totransform the “contrast” (or gradient) responses, Lsorf-adapted(Equation 11) to a luminance image. This method can becompared to previous filling-in used methods suggested formechanisms of assimilation effects (Grossberg, 2003; Speharet al., 2002). As far as we know, the previous used filling-inmethods, which relate to modeling psychophysical effects(such as assimilation and Kanizsa effects), do not enableapplication with real images. The modified Jacobi method(Quarteroni & Valli, 1994) used here (Equation 15) enablesthis application and with less arbitrary constrains. It has tobe mentioned that previous models which applied filling-incomponents used these components as part as the model. Inthe presented model, however, the filling-in component isused only as part of the inverse function in order totransform the “gradient” image to luminance image.The usage of the specific method for inverse function

leads to good image’s results, which are not stronglyinfluenced by the spatial resolution and are mildlyimpaired with the common simulated results such as with“halo” artifacts (Reinhard, Ward, Pattanaik, & Debevec,2005). Results images in this quality enable an evaluationof the predicted psychophysical effects (Figures 7–21).The quality level of output images can be compared tofew published predicted output images, mainly from thepredicted White’s effect done by ODOG and LDOGmodel in the computational studies (Blakeslee &McCourt, 2004; Robinson et al., 2007). Our results showimages with minimal distortions in the shape of thepredicted stimulus and predicted modulation in the testregions not less than in the surrounding region as observedin some predicted results in previous studies.Although the main reason of implementing the inverse

function was to evaluate the model performance, it doesnot exclude the possibility that such neuro-physiologicalprocess might exists. Such an application can beperformed by a simple neurological network consistingof several layers. This can be done by utilizing a parallelnetwork, containing lateral connections which has excita-tory connection to its surrounding neighboring neurons(the “surround” area in Equation 15) and also a feed-forward longitude excitatory connection (Lsorf-adapted inEquation 15).The model predicts also that higher spatial frequency or

smaller retinal size of the pattern contribute to yieldstronger assimilation effect (Figure 17). This crucialproperty is in agreement with many experimental studiesin assimilation effect (McCourt, 2005). This property ispredicted by model by yielding larger influence of the

contrast values in the remote area (Equation 6) to stimuliwith high spatial frequencies. In such stimuli, more edgesare present in the remote area than in the remote area oflow frequency stimuli. In addition, in a similar rational, themodel predicts larger assimilation effect while the contextarea contains larger number of “number of surround-bands” in the different assimilation stimuli for the samespatial frequency stimulus components (Bindman &Chubb, 2004) because more edges will be summed at theremote area.Psychophysical results showed (Spehar et al., 2002) that

the shift of the brightness is smaller at inverted White’seffect than in theWhite’s effect. We believe that this occurscommonly in psychophysical stimuli since in invertedstimuli usually the contrast ratio (between test andsurrounding) is smaller comparing to the classical effect.Spehar and colleagues (2002) address experimentally

the question regarding the differing views on whether theclassical and inverted White’s effects are mediated bycommon or separate underlying mechanisms. They variedthe aspect ratio of the test and inducing regions in theclassical and inverted White’s effects and compared twospecific sets of stimuli. They found that in the “portrait”orientation of the inverted White’s effect configuration(Figure 2 in Spehar et al., 2002) there is an oppositerelative brightness shift than the one they found in the“landscape” inverted White’s effect. Due to these exper-imental results (see Figure 3 in Spehar et al., 2002), theysuggested that induction in the classical and invertedWhite’s configurations originated from different under-lying mechanisms.We speculated that the contrast relation between their

type of stimuli (the Classical landscape and Classicalportrait) is different and expected to lead to oppositepredictions according to our model in relation to the testregion. Thus, this contrast difference can cause thedifferent trend of results. To test this assumption, we ranour computational model and found significant smallercontrast induction in the “portrait” stimuli (due to thesmaller spatial frequency). We speculated that the effect inthe “portrait” configurations is mainly mediated by theinduction of the first order. To test this assumption wecreated a “revised” version of the “classical portrait” withdecreased area of the black zones adjacent to the targetstimuli (Figure 2C in Spehar et al., 2002). In this “revised”configuration, the direction of the perceived brightness shiftis inverted. This brightness shift in the “revised” portraitconfiguration has also been predicted by the currentpresented model. Consequently, it supports the idea thatthe different trend of results (Figure 3 in Spehar et al.,2002) is derive from the induction of the first order(simultaneous contrast), which deals with brightnessinduction rather than contrast–contrast induction. Thecomputational analysis of this first-order induction is notin the scope of this current paper but was processedpreviously for the chromatic domain (Spitzer & Barkan,2005).

Figure 17. The same classical White’s effect and its invertedWhite’s effect are the same as in Figure 16, but with two timeshigher the spatial frequency (left column). The model’s results(right column and relevant curves) show that the perceived effectobtained is stronger for the configurations with higher spatialfrequency both for White’s and its inverted effect.

Journal of Vision (2008) 8(7):27, 1–26 Barkan, Spitzer, & Einav 19

Since the predicted results of both assimilation effectsand their inverted effects (Figures 7–13) are yielded fromthe very same model, we derive an opposite conclusion totheirs, i.e., that induction in the classical and invertedWhite’s configurations is due to the same underlyingmechanism. To further support these predictions we tested(Figure 16) the algorithm performance also on the original“landscape” (Figure 2 in Spehar et al., 2002), whichcontain a careful configuration which use the samecontrast range among the inducing stripes in both classicalassimilation and inverted assimilation effect. Figure 16shows the correct trend of results that is in agreement withthe perceived results.A further support to our model has been achieved by the

results to our scattered tests on the role of the spatialfrequency on the contrast–contrast effects. Figures 16 and 17presents the model’s results to two sizes of the “landscape”orientation of the classical White and the inverted Whiteeffects (from Figure 2 in Spehar et al., 2002). The resultsshow in this effect and additional assimilation stimuli (notpresented) that the higher the spatial frequency the largerthe assimilation and the inverted assimilations obtainedeffects, as perceived psychophysically.Several studies tried to predict through computational

models the two “contradicting” effects, simultaneouscontrast and several assimilation effects (Bressan, 2006;Howe, 2005; Robinson et al., 2007). We believe that thesetwo effects are ruled by two different mechanisms, theadaptation of the first order and the adaptation of the secondorder, correspondingly (Spitzer & Barkan, 2005). Theadaptation of the first order deals with perception of thelevels of brightness (or color) and the second order dealswith brightness (or color) contrast–contrast stimuli (tex-tures). The adaptation of the first order is processed in theretinal level and the second order is probably processed inthe cortical level (Spitzer & Barkan, 2005). Evidences ofexistence of these two effects can be seen in a clearer wayin chromatic stimuli that can manifest two differentchromatic directions, while in achromatic stimuli theperceived result is obtained in one domain that is the graylevel. The psychophysical results of colored White’s effectof Antis (see Figure 1.2 of Anstis, 2005) show the twodifferent chromatic directions toward complementary color(upward arrow) and toward embedded color (rightwardarrow). The same trend of results is obtained from ourprevious computational chromatic model of induction ofthe first and second orders. That model has been applied onreal images which show both color constancy (first-orderinduction) and chromatic contrast enhancement (second-order induction) (Figure 12 in Spitzer & Barkan, 2005).Due to this view on the two induction orders, we doexpect to get simultaneous contrast effects only from adap-tation of first order (which is not dealt within this currentbrightness model that emphasizes the contrast–contrastpredictions). This is a contrast to some other approachespresented in the literature, which expected to get botheffects in the very same model, as presented in ODOG

model (Blakeslee & McCourt, 1999, 2004; Robinson et al.,2007).Our proposed model does not contradict the suggested

idea that White’s effect is produced by a combination ofboth contrast and assimilation processes as suggested byBlakslee and McCourt (2004). Although only one mech-anism of adaptation of second order, contrast–contrastmechanism can predict the “classical” White’s effect andits inverted effect, as presented here, still in differentspatial arrangements additional mechanisms can play eventhe major role.In a following study we will present the predictions of

the two orders of brightness inductions as a compoundmodel, as we have shown for the two orders of chromaticinductions (Spitzer & Barkan, 2005). In any case an idealcomputational model is supposed to predict all config-urations of all effects that relates to the relevantmechanism. The presented model, however, presents onlythe effects that are refer only to the adaptation of thesecond order. Consequently, the presented model cannotpredict phenomena that are govern mainly by adaptationof the first order or other brightness mechanisms such asMach bands, Chevreul illusion, and Herman grid.In addition, the current model cannot account for all

spatial configurations of White’s effect. Depending alsoon the spatial scale White’s effect is produced by acombination of both contrast and assimilation processes(Blakeslee & McCourt, 2004). We believe that furtherresearch on the integration of first and second order ofadaptation might succeed in predicting the effect in allspatial configurations.Many of the studies mentioned the form of anchoring to

explain the assimilation effect or as an additional factorwhich contributes to the assimilation effect (Gilchrist et al.,1999). This anchoring model causes the lightest surface ina scene or an image to appear white. Such an explanationmight be useful for the inverted assimilation effect, but itcannot account for all configurations that exhibit theappearance of assimilation effect, such as the White’seffect (Economou, Zdravkovic, & Gilchrist, 2007). Inaddition, several studies reported that the anchoring modelfailed to predict additional assimilation effects (theinverted assimilation effect has not been referred to), suchas in the study of Bindman and Cubb (2004), which arealso presented here, Figures 14 and 15.In this study we suggest that assimilation and contrast–

contrast induction which includes both facilitation andsuppression derive from the very same mechanism. Thesame induction mechanism has to be studied for thechromatic corresponding effects. On the other hand, ourcurrent model (and other such as the ODOG model;Blakeslee & McCourt, 2004) is not suspected to predictthe grating contrast–contrast effects, as for example hasbeen demonstrated in D’Zmura and Singer (1998). Theissue of accounting the selectivity for orientation propertiesis beyond the scope of the current study and is planned tobe study by us in future.

Journal of Vision (2008) 8(7):27, 1–26 Barkan, Spitzer, & Einav 20

Figure 18. Illustration of the classical contrast–contrast effect (left column). The central gray squares in the left and right large squaresimages have identical contrast and luminance values. The appearance of the central squares is influenced by the surrounding contrastand this correct prediction’s effect is presented in the right column and the curve.

Journal of Vision (2008) 8(7):27, 1–26 Barkan, Spitzer, & Einav 21

Figure 19. An additional contrast–contrast effect (Spehar et al., 1996). The horizontal three uniform patches are identical in right and left sidesof the original image (left column). The model’s results image (right column) show the correct trend that is demonstrated in relevant curves.

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Figure 20. First and second-order White’s configurations. Figures 20 and 21 (left columns) termed in the literature as combination of firstand second-order White’s configurations (Spehar & Zaidi, 1997). Note that the average luminance of the inducing textures is identical;across, the two images and the luminance of the test stripes at each image share the same value. The result’s image (right column andthe relevant curves) shows the correct trend of prediction, for all the test stimuli.

Journal of Vision (2008) 8(7):27, 1–26 Barkan, Spitzer, & Einav 23

Figure 21. First and second-order White’s configurations on dark gray test stimuli. See description in Figure 20.

Journal of Vision (2008) 8(7):27, 1–26 Barkan, Spitzer, & Einav 24

Acknowledgments

We acknowledge Nir Sochen for his useful helpregarding the inverse function. We thank the AmanFoundation and the Adam Super Center for Brain Studiesfor funding the research.

Commercial relationships: none.Corresponding author: Hedva Spitzer.Email: hedva@eng.tau.ac.il.Address: Department of Biomedical Engineering, The Ibyand Aladar Fleischman Faculty of Engineering, Tel AvivUniversity, Ramat Aviv 69978, Israel.

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