Contributions of contour frequency, amplitude, andluminance to the watercolor effect estimated by conjointmeasurement

Peggy Gerardin # $

INSERM U846 Stem-Cell and Brain Research Institute,Department of Integrative Neurosciences, Bron, France

Universite Lyon 1, Lyon, France

Frederic Devinck # $Universite Rennes 2, CRPCC, Rennes, France

Michel Dojat # $INSERM U836, Universite Grenoble Alpes, GIN, Grenoble,

France

Kenneth Knoblauch # $

INSERM U846 Stem-Cell and Brain Research Institute,Department of Integrative Neurosciences, Bron, France

Universite Lyon 1, Lyon, France

The watercolor effect is a long-range, assimilative, filling-in phenomenon induced by a pair of distant, wavycontours of different chromaticities. Here, we measuredjoint influences of the contour frequency and amplitudeand the luminance of the interior contour on thestrength of the effect. Contour pairs, each enclosing acircular region, were presented with two of thedimensions varying independently across trials(luminance/frequency, luminance/amplitude, frequency/amplitude) in a conjoint measurement paradigm (Luce &Tukey, 1964). In each trial, observers judged which of thestimuli evoked the strongest fill-in color. Control stimuliwere identical except that the contours wereintertwined and generated little filling-in. Perceptualscales were estimated by a maximum likelihood method(Ho, Landy, & Maloney, 2008). An additive modelaccounted for the joint contributions of any pair ofdimensions. As shown previously using difference scaling(Devinck & Knoblauch, 2012), the strength increases withluminance of the interior contour. The strength of thephenomenon was nearly independent of the amplitudeof modulation of the contour but increased with itsfrequency up to an asymptotic level. On average, thestrength of the effect was similar along a givendimension regardless of the other dimension with whichit was paired, demonstrating consistency of theunderlying estimated perceptual scales.

Introduction

The watercolor effect (WCE), first described byPinna (1987), has provoked great interest over the lastdecade (Pinna, Brelstaff, & Spillmann, 2001). In itsclassic configuration, it is generated by two adjacentand thin, parallel, wavy contours of different chroma-ticity separating two regions in the visual field. Thecolor of one of the contours appears to diffuse into theregion it borders, filling it with a washed-out shade ofthe contour. The fill-in can occur over a large region ofthe visual field, raising the possibility that thephenomenon relates to high-level processes (Spillmann& Werner, 1996), such as surface perception (Pinna etal., 2001), figure-ground segregation (Pinna, Werner, &Spillmann, 2003; von der Heydt & Pierson, 2006; Pinna& Tanca, 2008; Tanca & Pinna, 2008), or other aspectsof perceptual organization (Pinna, 2005).

The strength of the WCE is influenced by severalfactors, including the relative luminances of thecontours (Devinck, Delahunt, Hardy, Spillmann, &Werner, 2006; Devinck & Knoblauch, 2012), the widthof the inducing contours (Pinna et al., 2001; Devinck,Gerardin, Dojat, & Knoblauch, 2014), and thecontinuity and contiguity of the contour pairs (Devinck& Spillmann, 2009). Most studies have assessed thestrength of the effect with matching or hue cancellation

Citation: Gerardin, P., Devinck, F., Dojat, M., & Knoblauch, K. (2014). Contributions of contour frequency, amplitude, andluminance to the watercolor effect estimated by conjoint measurement. Journal of Vision, 14(4):9, 1–15, http://www.journalofvision.org/content/14/4/9, doi:10.1167/14.4.9.

Journal of Vision (2014) 14(4):9, 1–14 1http://www.journalofvision.org/content/14/4/9

doi: 10 .1167 /14 .4 .9 ISSN 1534-7362 � 2014 ARVOReceived December 22, 2013; published April 10, 2014

techniques (Devinck, Delahunt, Hardy, Spillmann, &Werner, 2005; von der Heydt & Pierson, 2006). Morerecently, paired comparisons have been used toquantify the phenomenon. Cao, Yazdanbakhsh, andMingolla (2011) judged which of a pair of stimulishowed a more salient brightness filling-in for anachromatic version of the WCE. The strength of theWCE was related to the probability of distinguishingtwo patterns. Devinck and Knoblauch used maximumlikelihood difference scaling (MLDS) to estimate thestrength of the WCE as a function of the luminance ofthe interior inducing contour. MLDS involves com-parisons between large stimulus differences or intervalsand leads to an interval scale that describes changes instimulus appearance along a single dimension (Malo-ney & Yang, 2003; Knoblauch & Maloney, 2012b).

MLDS is a promising technique because it is basedon a Gaussian, equal-variance signal-detection model(Knoblauch & Maloney, 2012b). The estimated per-ceptual scales have been shown to predict discrimina-tion behavior, thus yielding an integrated description ofperformance from threshold to perception (Devinck &Knoblauch, 2012). A disadvantage is that the methodonly permits evaluation of one stimulus dimension at atime. This difficulty, however, is overcome by theparadigm of conjoint measurement (Luce & Tukey,1964; Krantz, Luce, Suppes, & Tversky, 1971; Fal-magne, 1985; Roberts, 1985; Knoblauch & Maloney,2012b), which permits estimating interval scales for twoor more dimensions from the same experimental data.In this paradigm, the observer judges one pair ofstimuli in each trial, but the levels of the dimensionsbeing studied vary independently across the two stimulifrom trial to trial. In this way, the influence onjudgments of varying one dimension while the other isfixed can be evaluated. Ho et al. (2008) recentlydeveloped a Gaussian, equal-variance, signal-detectionmodel of conjoint measurement and used maximumlikelihood to estimate the underlying perceptual scalesand to test several hypotheses on how observerscombine information across dimensions (maximumlikelihood conjoint measurement or MLCM). We usedthis procedure to study the joint influences ofluminance contrast, contour frequency, and amplitudeon the WCE.

Material and methods

Observers

Six observers (two male and four female) withnormal or corrected-to-normal vision volunteered forthe experiments (mean age 6 SD: 33 6 8 years). Threeparticipated in all three conditions, two in only one

condition, and one in two of the conditions. Allobservers but one (author PG) were naive, and all hadnormal color vision as assessed by a Farnsworth PanelD15. Observers who required optical corrections woretheir glasses while performing the experiments.

All experiments were in accordance with theprinciples of the Declaration of Helsinki.

Apparatus

The experiments were performed in a dark room.Stimuli were displayed on an Eizo FlexScan T562-Tcolor monitor (42 cm) driven by a MacBook Pro (2.2GHz). The screen had a resolution of 800 · 600 pixelsand was run at a field rate of 100 Hz, noninterlaced.The voltage-phosphor luminance relationship waslinearized with look-up tables. Calibration of the screenwas performed with a Minolta CS-100 chroma meter.Observers were placed at a distance of 57.3 cm from thescreen.

Stimuli

Each stimulus was generated with Matlab R2009b(http://www.mathworks.com/) and displayed with thePsychophysics Toolbox extensions (Brainard, 1997;Pelli, 1997). All stimuli were displayed on a whitebackground (128 cd/m2, CIE xy ¼ 0.29, 0.30). Thecontour pairs that defined the stimuli were each ofwidth 16 min, that is, 8 min for the interior and exteriorcontours, each. The outer contour was purple (CIE xy¼ 0.32, 0.19) and the inner orange (CIE xy¼0.48, 0.34).The stimuli were also specified in the DKL color space(Derrington, Krauskopf, & Lennie, 1984) with purpleand orange contours at azimuths of 3208 and 458,respectively. Control stimuli were identical except thatthe contours were interlaced and generated little filling-in (Figure 1c).

For independent control of the frequency and theamplitude of the WCE contours, they were constructedas Fourier descriptors (Zahn & Roskies, 1972). Eachstimulus was defined by a circle of 48 diameter whoseradius was modulated sinusoidally as a function ofangle according to the equation

RðhÞ ¼ rþ Asinð2pfhÞ; ð1Þwhere R is the stimulus radius at angle h, r the averageradius of the stimulus, A the modulation, and f thefrequency in cycles per revolution (cpr). The stimuli arethen plotted by transforming the polar coordinates (R,h) to rectangular coordinates (x, y) with the equations

xðR; hÞ ¼ RðhÞsinð2phÞ ð2Þ

yðR; hÞ ¼ RðhÞcosð2phÞ: ð3Þ

Journal of Vision (2014) 14(4):9, 1–14 Gerardin, Devinck, Dojat, & Knoblauch 2

For each condition, a factorial design was defined inwhich five levels along each of two dimensions(frequency, amplitude, or luminance) were chosen, andall levels crossed, creating a table of 25 stimuli. Figure1a shows an example for frequency and amplitude withfrequency varying across columns and amplitude acrossrows. The five frequencies were equally spaced, rangingfrom 4 to 20 cpr. The five amplitudes evaluated rangedfrom 0.04 to 0.2, and the five luminances of the orangecontour ranged from 0.1 to 0.9 equally spaced inelevation in DKL space. In two other conditions, thedimensions of frequency/luminance and amplitude/luminance were similarly crossed.

Procedure

Observers were run in three conditions (luminance/frequency, luminance/amplitude, frequency/amplitude)with a single condition tested within a session. In eachtrial, a pair of different stimuli from the stimulus tablefor that condition (e.g., Figure 1a) was chosen atrandom and presented to the observer. Observersjudged which of the two patterns evoked the mostsalient fill-in color. To test that observers respondedaccording to the fill-in appearance and not on the basis

of some other stimulus feature(s), an equal number oftest and control stimuli were interleaved in each session(e.g., Figure 1b and c). There are 25 · 24/2¼ 300 suchpairs. In each trial, either a pair of test or controlstimuli were presented. A session consisted of a randompresentation of all 600 test and control pairs. Eachcondition was repeated in five sessions, yielding 1,500test and 1,500 control trials per condition for a givenobserver.

Model

We modeled the data following the frameworkdescribed by Ho et al. (2008) for analyzing conjointmeasurement experiments (see also chapter 8, Kno-blauch & Maloney, 2012b). They define three nestedmodels that can be fit to describe the choices of theobservers in order of complexity: an independencemodel, an additive model, and a saturated model. Webegin the description with the additive model and thendescribe the simpler and more complex models withrespect to it.

We represent the stimulus levels along the twodimensions of a condition by a variable /ij, i, j¼ 1, . . .5, where the indices refer to two of the dimensions in

Figure 1. (a) Example of stimulus set for a conjoint measurement experiment. The table shows the set of stimuli used in the

frequency/amplitude condition. Each column corresponds to a different frequency and each row to a different amplitude. (b) In each

trial, a pair of stimuli were chosen from the table. The test stimuli contained a continuous interior orange contour. (c) For the control

stimuli, the interior contour was braided with the exterior contour. Observers were asked to judge which member of the pair evoked

the most salient filling-in.

Journal of Vision (2014) 14(4):9, 1–14 Gerardin, Devinck, Dojat, & Knoblauch 3

the set (luminance, frequency, amplitude), for example,a row and column, respectively, of Figure 1a. In theadditive model, we suppose that each of the twodimensions contributes to a filling-in response, w1

i , w2j ,

respectively, which depends on its physical intensitylevel, and that the filling-in response, wij, to a stimulus,/ij, is the sum of the component responses

wij ¼ w1i þ w2

j : ð4Þ

We assume that the observer judges the stimulus /ij

to display stronger filling-in than the stimulus /kl when

dijkl ¼ wij � wkl ¼ ðw1i þ w2

j Þ � ðw1k þ w2

l Þ. 0: ð5Þ

This representation yields an interval scale (Krantzet al., 1971). Note, for example, that by rearrangingEquation 5 as follows,

dijkl ¼ ðw1i � w1

kÞ � ðw2l � w2

j Þ. 0; ð6Þ

the decision rule is equivalent to a comparison ofintervals across dimensions.

It is unlikely, however, that the observer would makethe same response in every trial when the two stimuliare very similar. To incorporate this inherent variabilityof human responses, we suppose that the decisionvariable, Dijkl, is contaminated by internal noise so thatthe observer chooses the first stimulus exactly when

Dijkl ¼ dijkl þ �ijkl . 0; ð7Þwhere each �ijkl is a draw from a distribution ofindependent and identically distributed normal vari-ables with l ¼ 0 and variance¼ 4r2. This is an equal-variance, Gaussian signal-detection model. The coeffi-cient of four on the variance parameterizes theestimated scale values so that variance of the responsealong each dimension is equal to r2. As a result, theestimated response values wk

i /r are distributed asnormal variables with r2¼ 1 and are, thus, on the samescale as the sensitivity measure d0 from signal detectiontheory (Green & Swets, 1966).1

With p intensity levels sampled along each dimensionand the estimate of the variance of �, there are 2pþ 1parameters in the model. The reparameterizations ofthe scale described above suggest that we may multiplythe estimated values by any constant without changingthe model predictions of the observer’s responses. Infact, we may similarly add any constant to the scaleswith similar results. To fix the estimated scales, we setthe lowest value of each scale to zero.

This eliminates two parameters from the fit. Inaddition, by making r2¼ 1, we eliminate one more sothat the fitting process requires estimating 2p – 2parameters. The parameterization that we propose isslightly different from that used by Ho et al. (2008),who estimated r but normalized the estimatedfunctions so that the maximum scale value was unity.

Both parameterizations yield identical predictions ofperformance.

The additive model implies that there are componentfunctions representing the internal response to eachphysical dimension and that the overall response to astimulus is the simple sum of these componentresponses. We illustrate this in Figure 2c and d. InFigure 2c, we show two hypothetical componentresponses, w1 and w2, each of which, for simplicity, is alinear function of the stimulus level. Stimulus level hereand elsewhere is indicated by an index, not physicalunits. This convention, as used elsewhere (Ho et al.,2008; Knoblauch & Maloney, 2012b), allows the scalesfor both dimensions to be plotted together. Theresponse to any stimulus with levels i and j, respec-tively, along the two stimulus dimensions is representedby the sum of their component responses. As anexample, consider a stimulus of level three along thefirst dimension and level four along the second. Thecomponent responses to the level along each dimensionare indicated by the two black points. Figure 2d showsthe set of summed responses for all pairings of the twodimensions with the base dimension corresponding tow2 and the parameter indicated along each curvecorresponding to w1. Each curve has the shape of thecomponent curve for w2 but is displaced vertically bythe value of w1, resulting in a set of parallel contours.Analogous to a linear model, we could say that eachdimension shows a main effect but no interaction. Thesummed response to the stimulus with the levels.indicated in Figure 2c is indicated by the black point.

A simpler model than the additive model is obtainedby assuming that the observer’s judgments depend ononly one of the component dimensions. In that case,the decision variable reduces to

Dijkl ¼ w1i � w1

k þ �ijkl ð8Þwhere we assume that only the first dimensioncontributes to the judgments. Ho et al. (2008) call thisthe independent-property model, but we will refer to itas just the independence model. The independencemodel requires estimating only p – 1 parameters.

The independence model is illustrated in Figure 2aand b. In Figure 2a, the component w1 increases withstimulus level along the first dimension, but w2 is flat orindependent of the level of the second dimension. Thesummed responses are indicated in Figure 2b by a set oflines of zero slope that are vertically displaced by theresponses along the w1 curve. In analogy to a linearmodel, there is a significant main effect of the firstdimension but not of the second.

Finally, the additive model may not suffice toprovide a satisfactory description of the observer’sresponses. Ho et al. (2008) describe the full model,which includes interaction terms, w12

ij , that depend onthe intensity levels of both dimensions. The decision

Journal of Vision (2014) 14(4):9, 1–14 Gerardin, Devinck, Dojat, & Knoblauch 4

variable is

Dijkl ¼ ðw1i þ w2

j þ w12ij Þ � ðw1

k þ w2l þ w12

kl Þ þ eijkl:

ð9ÞThis model requires the estimation of p2 – 1

parameters, one less than the number of stimuli tested,and will be referred to as the saturated model. Anexample of response curves that might result from asaturated model are shown in Figure 2e. The importantfeature is that the curves are not parallel and, thus, thecurves cannot be explained by a simple additivecombination of component curves as in the previoustwo cases.

Analyses

The three models described above form a nestedsequence. The estimated scale values with the aboveconstraints that best predict the observers’ responsesare fitted by maximum likelihood (Ho et al., 2008).Because the decision variable is linear in the estimatedcoefficients, however, the fitting procedure is simplyimplemented as a generalized linear model with a

binomial family (McCullagh & Nelder, 1989). Standarderrors and confidence intervals of estimated scale valuesfor comparing model fits were estimated using abootstrap procedure (Knoblauch & Maloney, 2012a,2012b). All analyses were performed using the opensource software R (R Development Core Team, 2012)with functions from the MLCM package (Knoblauch& Maloney, 2012a, 2012b).

Results

Judgments from the observers for test and controlconditions for each of the pairings of the threedimensions are shown in Figure 3a through c in aformat introduced by Ho et al. (2008) and termed aconjoint proportion plot (CPP) by Knoblauch andMaloney (2012a). Each CPP summarizes the propor-tion of times that the ordinate stimulus, Skl, was judgedto show a greater fill-in than the abscissa stimulus, Sij,coded according to the grey levels shown in the colorbar at the right of each set of graphs, for every stimuluspair presented. The levels along the two dimensions are

Figure 2. Schematic examples of the three models. (a) Illustrative component curves for an independence model. (b) Predicted

responses for each combination of stimulus attributes for the independence model. (c) Illustrative component curves for the additive

model. (d) Predicted responses for each combination of stimulus attributes for the additive model. (e) Predicted responses for a

saturated model. Filled black points in c and d are described in the text.

Journal of Vision (2014) 14(4):9, 1–14 Gerardin, Devinck, Dojat, & Knoblauch 5

represented along each axis using a scheme in which a 5· 5 outer grid demarcates the stimulus levels along onedimension (e.g., amplitude in Figure 3a), and the levelsof the second dimension are represented as an innersubgrid nested within each square of the outer grid(e.g., frequency in the same figure). As the responses arecombined across random left/right orders in thepresentation—only the upper left triangle of a CPP isunique and displayed.

In Figure 3a through c, the set of results from eachobserver is indicated by his/her initials above the pairof CPPs for test (upper) and control (lower) stimuli.Figure 3d shows the expected pattern of responses foran ideal observer for whom the judgments are based onthe level of only one of the two stimulus dimensions.The left CPP shows the results when the judgmentsdepend solely on the outer dimensions (Dim 1) and theright on the inner (Dim 2). In general, the CPP for thetest stimuli resemble more closely the ideal patterndisplayed for the inner dimension, suggesting that thecontribution of frequency dominated when bothfrequency and amplitude covaried but that luminancecontributed more strongly to the judgments when

paired with the other two dimensions. The patterns do,however, deviate from the ideal, indicating contribu-tions from both dimensions.

There is an important difference in the procedureused here from that reported by Ho et al. (2008). Theytested stimuli that varied along the dimensions ofroughness and glossiness and asked different observersto judge which member of a pair appeared rougher orshinier. Thereby, they could estimate the contamina-tion of the one dimension to judgments along the other.In our experiments, the task is not necessarily related tothe stimulus dimensions. Thus, one cannot be sure apriori that the observer is actually basing his/herjudgments on the stated criterion. Instead, the observermay simply be judging the variation along the twodimensions manipulated, that is, which stimulus has ahigher frequency contour, amplitude, or luminance. Ifobservers were performing in this manner, then wewould expect to see the same pattern of responses in theCPPs for control and test stimuli. By and large,however, the patterns are quite different with thecontrol CPP patterns appearing to be random. Theprincipal exception to this is evident in the control data

Figure 3. Conjoint proportion plots. Each plot shows the proportion of stimulus, Skl, judged to show a greater fill-in than the stimulus

represented on the abscissa, Sij, as grey level coded according to the color bar at the right. Levels i and k along one dimension are

indicated by the exterior axis numbers. Each of the squares (i, k) is subdivided in a 5 · 5 grid indicating the pairings (j, l) along the

second dimension. Interior axis labels are shown for the extreme levels in the two lower right squares of each plot. In each set of

plots, the top row indicates the results for the test stimuli and the bottom for the control with observers’ initials indicated in the

upper strips. (a) Outer/inner dimensions: ampitude/frequency. (b) Amplitude/luminance. (c) Frequency/luminance. (d) Expected

response patterns for an observer who judges the stimuli based on the contributions along only one of the dimensions. Dim 1 shows

the expected pattern for the outer dimension and Dim 2 for the inner dimension.

Journal of Vision (2014) 14(4):9, 1–14 Gerardin, Devinck, Dojat, & Knoblauch 6

of observer OH for the amplitude/luminance condition(Figure 3b) and, to a lesser extent, for observer DL inthe same condition. In these two cases, then, we cannotexclude that either the observers perceived a smallamount of filling-in or, alternatively, that they showeda tendency to base their judgments on the physicaldimensions of the stimulus for a subset of the trialsrather than on the perceived filling-in per se.

Figure 4a through c shows the average estimatedscales under the additive model for each observer foreach pairing of frequency, luminance, and amplitude. Adifferent color is used to indicate the contributions tothe judgment of each dimension: red¼ frequency, blue¼ amplitude, green¼ luminance. The top row of graphsin each subfigure shows the scale values estimated forthe test stimulus and the bottom row for the control.

For example, Figure 4a indicates that the contributionof the frequency dimension to the filling-in strengthincreases with the frequency of the contour while theamplitude dimension shows a smaller effect thatincreases between the first and second levels tested forall observers but one (SJ shows a relatively flatamplitude contribution) and then remains approxi-mately constant. These results contrast with the scalesestimated for the control stimuli that show nodifference and very little effect of either dimension onthe strength of filling-in.

Similarly, Figure 4b shows that the contribution ofluminance to the filling-in increases with the luminanceof the orange contour while the amplitude contribu-tions are similar to those obtained in Figure 4a with theexception of observer SJ who shows a stronger

Figure 4. (a) Results for Experiment 1. Additive model average estimates for perceived WCE for test and control patterns as a function

of frequency (red circles) and amplitude (blue circles) of the contour for four observers. Means of five sessions; error bars are 95%

confidence intervals. (b) Results for Experiment 2. Additive model average estimates for perceived WCE for test and control patterns

as a function of luminance (green circles) and amplitude (blue circles) of the contour for four observers. Means of five sessions, error

bars are 95% confidence intervals. (c) Results for Experiment 3. Additive model average estimates for perceived WCE for test and

control patterns as a function of luminance (green circles) and frequency (red circles) of the contour for four observers. Means of five

sessions; error bars are 95% confidence intervals. (d) Mean estimated scales for each pairing of dimensions. Error bars indicate 95%

confidence intervals for observer differences. Color codes correspond to those used in previous three figures.

Journal of Vision (2014) 14(4):9, 1–14 Gerardin, Devinck, Dojat, & Knoblauch 7

luminance contribution than in the previous condition.The controls show little influence of either dimension.The exception is again shown by observer OH,consistent with the CPP plots in Figure 3b. Note also,that, for this condition, the responses of observers PGand OH to the test stimuli result mostly from theluminance dimension, again, as suggested by compar-ison of their CPP plots to the ideal cases.

Figure 4c shows the contributions of luminance andfrequency when these two dimensions are varied. Theluminance scales are similar to those in Figure 4b, andthe frequency scales resemble those obtained in Figure4a. As in the preceding figures, the control stimuliprovide little evidence for filling-in in that thecontributions of each dimension to the judgmentsremain low as the scale value is increased.

Finally, Figure 4d shows the means and 95%confidence intervals across observers for each pairing ofthe three dimensions. The scales for the controlconditions show little effect with confidence intervalsoverlapping zero, and confidence intervals for testconditions exclude zero with the exception of thehighest amplitude in the amplitude/luminance condi-tion. The large confidence intervals for some of the testconditions reflect the individual differences shown inthe previous graphs. Interestingly, the average scale foreach dimension tends to be similar, independently ofthe other dimension with which it is paired, a point wewill examine in more detail below.

Comparisons of independence and additivemodels

We consider the possibility that the contribution ofonly one dimension independently of the other sufficesto account for observers’ judgments, that is, theindependence model. We chose to model the data basedon the dimension that generated the strongest contri-butions under the additive model. We will refer to thisdimension as the primary dimension and the other asthe secondary dimension. Specifically, luminance wasthe primary dimension when paired with frequency oramplitude, and frequency was the primary dimensionwhen paired with amplitude.

We compare the models by plotting the estimatedscales from both models in the same graph for eachobserver. For reference, we are comparing the twomodels shown in Figure 2a and c. Comparisionsbetween additive (solid) and independence (greydashed) model fits are displayed in Figure 5 with 95%confidence intervals of the additive estimates displayed.By construction, the contribution of the secondarydimension is fixed at zero for the independence model.

The comparisons in Figure 5 indicate that scaleestimates of the primary dimension under the inde-

pendent model tend to correspond closely to thoseunder the additive model. For observer SJ in theamplitude/luminance condition and for all observers inthe frequency/luminance condition, however, the esti-mate of the primary dimension contribution for theindependence model falls outside the confidence limitsfor the additive model. This is sufficient to reject theindependence model for these conditions. Moreover, inall cases but one (observer SJ for the amplitude/frequency experiment), the secondary dimension of theadditive model differs significantly from zero, theprediction along the secondary dimension for inde-pendence, thus leading to rejection of the independencemodel in nearly every case. Thus, although thecontribution of amplitude to the judgments is small, itis significant.

Comparisons of additive and saturated models

To compare the additive and saturated model fits, weadopt the format of Figure 2d and e in which theestimated scale values are plotted as a function of thestimulus index along one dimension with the otherdimension treated as a parameter. The comparisons areshown for each pair of dimensions and for eachobserver in the panels of Figure 6. The results arenoisier than for the preceding comparison. However,for most of the comparisons, the confidence intervalsfor the saturated model (solid) overlap the curves forthe additive model (dashed), suggesting that the simpleradditive model suffices to describe the data. Most of theexceptions concern the amplitude dimension that, as wesaw above, only weakly contributes to the judgmentscompared to the frequency or the luminance anddisplays more variability across conditions. In addition,the deviations that occur are not systematic acrossobservers.

Residual differences between the two models areshown as grey level plots in Figure 7a through c. Thedifferences have been normalized with respect to thestandard errors of the saturated model fits. In general,there is no obvious pattern in the residuals although thefrequency/amplitude condition appears noisier than theother two. For this condition, eight of the 125 residuals(6.4%) exceed a magnitude of two whereas none of theresiduals exceed a value of two for the other twoconditions.

Comparisons of estimated scales acrossexperiments

To what extent are the estimated contributions fromeach dimension invariant with respect to the context ofthe judgments? For example, does the estimated scale

Journal of Vision (2014) 14(4):9, 1–14 Gerardin, Devinck, Dojat, & Knoblauch 8

for the contribution of luminance depend on whethereither frequency or amplitude is the covarying dimen-sion? The same question can be posed for eachdimension with respect to the other two. Figure 8ashows the estimated contribution of luminance to thefill-in color when the contour frequency (red) or itsamplitude (blue) is the covarying dimension as afunction of the luminance index for the four observerswho performed both experiments. The scales obtainedunder the different conditions agree very well for all ofthe observers, and the overlap in the confidenceintervals lends no support to the importance of thedifferences in the means.

Similarly, Figure 8b and c shows the comparisons ofthe scales obtained for the frequency and amplitudedimensions, respectively. There is good agreementacross conditions for the frequency scales for observersPG and SJ. For OH, the scales agree in form althoughthe sensitivity to frequency is higher when paired withthe luminance than with the amplitude dimension.

There seems to be less agreement between the scalesbased on the amplitude dimension. This is mostmarked for observer SJ, for whom the contribution ofamplitude to the judgments is insignificant whenpaired with frequency but significantly higher, basedon the nonoverlap of confidence intervals, whenpaired with the luminance dimension. As noted

Figure 5. Comparisons of additive and independent model fits for each observer in the three experiments. The solid lines indicate the

estimated contributions of each dimension under the additive model, and the dashed grey lines indicate the estimates under the

independence model. The color codes correspond to those of the points from Figure 4 with red¼ frequency, green¼ luminance, and

blue ¼ amplitude. The error bars are bootstrap estimated 95% confidence intervals (n ¼ 10,000) under the additive model.

(a) Amplitude/frequency. (b) Amplitude/luminance. (c) Frequency/luminance.

Journal of Vision (2014) 14(4):9, 1–14 Gerardin, Devinck, Dojat, & Knoblauch 9

previously (see Figure 4a and b), however, theamplitude contributions tended to be small and morevariable. This is borne out by the averages acrossobservers shown in Figure 8d that show goodagreement in the scales measured along each dimen-sion and agrees with the averages based on allobservers from Figure 4d. The average data show theamplitude contribution asymptotes by the secondstimulus level, which corresponds to an amplitude of0.08. The frequency contribution asymptotes by thethird level, which corresponds to a value of 12 cpr.

Discussion

We used the MLCM technique to quantify thecontributions to the filling-in strength of the WCE ofthree stimulus dimensions. An important differencefrom the previous use of this technique by Ho et al.(2008) is that the perceptual judgment required of theobservers in our experiments is not directly linked toeither of the stimulus dimensions whereas in Ho et al.(2008) it was. Thus, in theory, the observers could have

Figure 6. Comparisons of additive and saturated model fits for each observer in the three experiments. The solid lines indicate the

estimated contributions for each combination of the two dimensions under the saturated model, and the dashed lines indicate the

contributions under the additive model. Color is used to code the index of the secondary dimension (black, red, blue, green, and

gray for indices 1 to 5, respectively). The error bars are bootstrap estimated 95% confidence intervals (n ¼ 10,000) under the

saturated model. (a) The estimated contributions as a function of the index of the frequency variable for each amplitude tested.

(b) Estimated contributions as in panel (a) but as a function of the amplitude index with the curves parameterized by luminance.

(c) The estimated contributions as in the previous panels but as a function of the frequency index with the curves parameterized by

luminance.

Journal of Vision (2014) 14(4):9, 1–14 Gerardin, Devinck, Dojat, & Knoblauch 10

simply based their judgments directly on the stimulusdimensions manipulated (luminance, frequency, and/oramplitude) rather than the perceptual criterion offilling-in strength. To evaluate this possibility, weincluded control stimuli during the sessions for whichvery little, if any, filling-in was expected to occur. If theobservers based their judgments on the manipulateddimensions rather than the perceived filling-in, then theestimated contributions to the judgments for controland test stimuli would be the same. In fact, the modelestimates for the control conditions were severelyattenuated in all conditions, thus supporting that thetest conditions do reflect judgments with respect tofilling-in.

As shown previously using a difference scalingtechnique (Devinck & Knoblauch, 2012), the strengthof the WCE depends importantly on the luminance ofthe interior contour. We show here, as well, that the

filling-in is stronger for higher frequencies of thecontour modulation although the influence of contourfrequency asymptotes at about 12 cpr. Probablybecause of its small value, the estimates of thecontribution of amplitude to the phenomenon werefound to be more variable across conditions andobservers. In addition, the results are best supported bya model in which the responses to the separatedimensions contribute in an additive fashion.

It has previously been reported that the WCE ispresent although weaker with straight contours (Pinnaet al., 2001). Our results confirm this observation inthat, for nearly all conditions, the amplitude contribu-tion was greater for higher amplitudes than for thelowest value tested. We did not test a frequency of zerocpr or a zero amplitude because it would have led to asingular set of conditions, a row and column ofidentical stimuli in the frequency/amplitude matrix of

Figure 7. Residual differences between additive and saturated model fits shown in Figure 6 for all observers and conditions plotted as

grey levels according to color bars at the right of each plot. (a) Residuals for model fits of frequency/amplitude pairing for each

observer. (b) Residuals for models fits of amplitude/luminance pairing. (c) Residuals for models fits of frequency/luminance pairing.

The residuals have been normalized with respect to the estimated standard errors for the saturated model fits.

Journal of Vision (2014) 14(4):9, 1–14 Gerardin, Devinck, Dojat, & Knoblauch 11

stimuli (Figure 1). Additionally, because we fix thelowest perceptual scale values at zero in order to makethe estimated coefficients identifiable, our measure-ments of the strength of the phenomenon are only inreference to this value. A measure of the strength at thislowest value could have been obtained by includingpaired comparisons between test and control stimuli inthe experiment, but this would have quadrupled thenumber of stimulus pairs to evaluate.

We performed the experiments analyzing dimensionspairwise, that is, two-way experiments. Following thelogic of factorial design, it might be argued that itwould be more efficient to perform an experiment withall three attributes permitted to covary simultaneously,that is, a three-way experiment. Employing five levelsalong each dimension, as here, would generate 53¼ 125different stimuli and 125 · 124/2¼ 7,750 pairedcomparisons, which could be reasonably argued isexcessive. Simulation results for two-way experimentssuggest that it is the number of total judgments and notthe number of conditions that determines the precision

of the estimates (Knoblauch & Maloney, 2012b).Similar results have been obtained for the MLDStechnique (Maloney & Yang, 2003). This raises thepossibility of obtaining good scale estimates for a three-way experiment by subsampling from the full set ofstimulus pairs although it will be necessary to verifysuch a conjecture via simulation.

Comparing the dimensions pairwise permitted us toevaluate whether the estimated contributions of eachdimension to the observer’s judgments were indepen-dent of the contributions of the second, covaryingdimension. The results support such an invariance forthe luminance and frequency dimensions but are moreequivocal for the amplitude although all three dimen-sions are consistent with invariance for the averagedata. We suspect that the failures of invariance alongthe amplitude dimension stem more from its weakcontribution to the WCE, rendering it difficult toestimate with the number of trials that we employedand possible criterion shifts of the observers. Thesehypotheses remain to be tested, however.

Figure 8. (a) Comparison of the average perceptual scales as a function of contour luminance for four observers. Red symbols

correspond to the experiment frequency · luminance and blue symbols to the experiment amplitude · luminance. Each point is the

mean of five sessions, and the error bars are 95% confidence intervals. (b) Comparison of the average perceptual scales as a function

of contour frequency for three observers. Green symbols correspond to the experiment frequency · luminance and blue symbols to

the experiment amplitude · frequency. Each point is the mean of five sessions, and the error bars are 95% confidence intervals.

(c) Comparison of the average perceptual scales as a function of contour amplitude for three observers. Red symbols correspond to

the experiment amplitude · luminance and green symbols to the experiment amplitude · frequency. Each point is the mean of five

sessions, and the error bars are 95% confidence intervals. (d) Mean comparison of the average perceptual scales as a function of the

indices of contour amplitude (left, n¼ 3), of contour frequency (middle, n¼ 3), and of contour luminance (right, n¼ 4). Blue symbols

correspond to experiments with amplitude, green symbols to experiments with luminance, and red symbols to experiments with

frequency. The error bars are 95% confidence intervals based on interindividual differences.

Journal of Vision (2014) 14(4):9, 1–14 Gerardin, Devinck, Dojat, & Knoblauch 12

The WCE depends on both local spatial andchromatic properties of the contour of the stimulus butgenerates a long-range filling-in percept as well as afigure-ground segregation between the fill-in region andthe surround. Models to account for the WCE,therefore, typically include multiple levels of processing(Pinna et al., 2001; Pinna & Grossberg, 2005; von derHeydt & Pierson, 2006). We have recently shown(Devinck et al., 2014), for example, that the dependenceof the strength of the WCE on the spatial frequency ofthe inducing contour width has a narrow tuning similarto a class of luminance-chromatic cells found in corticalarea V1 of the macaque (Shapley & Hawken, 2011).Nevertheless, the spatial extent of V1 receptive fieldsand their interactions would seem to preclude this siteas the substrate for the long-range filling-in phenom-enon. Sensitivity to the frequency of the contours alsoseems unlikely to be a property of area V1 becauseselectivity for contour curvature is only observed inhigher-order visual areas (Schwartz, Desimone, Al-bright, & Gross, 1983; Hegde & Van Essen, 2000; Roeet al., 2012) and modeled as such (Habak, Wilkinson,Zakher, & Wilson, 2004; Poirier & Wilson, 2006). Ourresults do not distinguish directly between a mechanismbroadly tuned to the contour frequency and anensemble response over individual elements that arenarrowly tuned. The fact that the additive modeladequately explains the results, however, can be takenas consistent with a mechanism whose response isinvariant for a trade-off of the response along onedimension with respect to that along another; that is,the response of increase along one dimension could becompensated by decrease along the other. Such aninterpretation would favor a single mechanism withbroad tuning to curvature.

Keywords: psychophysics, scaling, conjoint measure-ment, filling-in, color vision, assimilation, watercoloreffect

Acknowledgments

This work was supported by a grant, ANR11JSH20021, from the Agence Nationale de la Re-cherche to Frederic Devinck. John S. Werner isgratefully thanked for comments.

Commercial relationships: none.Corresponding author: Kenneth Knoblauch.Email: ken.knoblauch@inserm.fr.Address: INSERM U846 Stem-Cell and Brain Re-search Institute, Department of Integrative Neurosci-ences, Bron, France and Universite Lyon 1, Lyon,France.

Footnote

1See Devinck and Knoblauch (2012) and Knoblauchand Maloney (2012b), p. 202, for the derivation forMLDS and Devinck and Knoblauch (2012) forempirical verification. The MLCM model is formallythe same as that for MLDS except for two sign changesin the decision variable, and therefore, the same ideasshould apply.

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