an equivalent illuminant model for the effect of surface...

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An Equivalent Illuminant Model for the Effect of SurfaceSlant on Perceived Lightness

M. BlojDepartment of Optometry, University of Bradford

Bradford, UK

C. RipamontiDepartment of Psychology, University of Pennsylvania Philadelphia, PA,

USA

K. MithaDepartment of Psychology, University of Pennsylvania Philadelphia, PA,

USA

R. HauckDepartment of Psychology, University of Pennsylvania Philadelphia, PA,

USA

S. GreenwaldDepartment of Psychology, University of Pennsylvania Philadelphia, PA,

USA

D. H. BrainardDepartment of Psychology, University of Pennsylvania Philadelphia, PA,

USA

Keywords: computational vision, equivalent illuminant, lightness constancy, scene geometry, surface slant

AbstractIn the companion paper (Ripamonti

et al., 2004), we present data thatmeasure the effect of surface slant onperceived lightness. Observers areneither perfectly lightness constantnor luminance matchers, and there isconsiderable individual variation inperformance. This paper develops aparametric model that accounts for howeach observer’s lightness matches varyas a function of surface slant. Themodel is derived from consideration ofan inverse optics calculation thatcould achieve constancy. The inverseoptics calculation begins withparameters that describe theillumination geometry. If theseparameters match those of the physicalscene, the calculation achievesconstancy. Deviations in the model’sparameters from those of the scenepredict deviations from constancy. Weused numerical search to fit the modelto each observer’s data. The modelaccounts for the diverse range ofresults seen in the experimental datain a unified manner, and examinationof its parameters allows

interpretation of the data that goesbeyond what is possible with the rawdata alone.

IntroductionIn the companion paper (Ripamonti

et al., 2004), we report measurementsof how perceived surface lightnessvaries with surface slant. The dataindicate that observers take geometryinto account when they judge surfacelightness, but that there are largeindividual differences. This paperdevelops a quantitative model of ourdata. The model is derived from ananalysis of the physics of imageformation and of the computations thatthe visual system would have toperform to achieve lightnessconstancy. The model allows forfailures of lightness constancy bysupposing that observers do notperfectly estimate the lightinggeometry. Individual variation isaccounted for within the model byparameters that describe eachobserver’s representation of thatgeometry.

Figure 1 replots experimental datafor three observers (HWK, EEP, and

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FGS) from Ripamonti et al. (2004).Observers matched the lightness of astandard object to a palette oflightness samples, as a function ofthe slant of the standard object. Thedata consist of the normalizedrelative match reflectance at eachslant. If the observer had beenperfectly lightness constant, the datawould fall along a horizontal line,indicated in the plot by the reddashed line. If the observer weremaking matches by equating thereflected luminance from the standardand palette sample, the data wouldfall along the blue dashed curvesshown in the figure. The completedata set demonstrates reliableindividual differences ranging fromluminance matches (e.g. HWK) towardsapproximations of constancy (e.g.FGS). Most of the observers, though,showed intermediate performance (e.g.EEP).

Figure 1 about here

Figure 1. Normalized relative matches, replotted fromRipamonti et al. (2004). Data are for Observer HWK (Paintinstructions), Observer EEP (Neutral instructions), andObserver FGS (Neutral instructions). See companion paper forexperimental details.

Given that observers are neitherperfectly lightness constant norluminance matchers, our goal is todevelop a parametric model that canaccount for how each observer’smatches vary as a function of slant.Establishing such a model offersseveral advantages. First, individualvariability may be interpreted interms of variation in modelparameters, rather than in terms ofthe raw data. Second, once aparametric model is established, onecan study how variations in the sceneaffect the model parameters (c.f.Krantz, 1968; Brainard & Wandell,1992)). Ultimately, the goal is todevelop a theory that allowsprediction of lightness matches acrossa wide range of scene geometries.

A number of broad approaches havebeen used to guide the formulation ofquantitative models of contexteffects. Helmholtz (Helmholtz, 1896)suggested that perception should be

conceived of as a constructedrepresentation of physical reality,with the goal of the constructionbeing to produce stablerepresentations of object properties.The modern instantiation of this ideais often referred to as thecomputational approach tounderstanding vision (Marr, 1982;Landy & Movshon, 1991). Under thisview, perception is difficult becausemultiple scene configurations can leadto the same retinal image. In thecase of lightness constancy, theambiguity arises because illuminantintensity and surface reflectance cantrade off to leave the intensity ofreflected light unchanged.

Because the retinal image isambiguous, what we see depends notonly on the scene but also on therules the visual system employs tointerpret the image. Various authorschoose to formulate the these rules indifferent ways, with some focusing onconstraints imposed by knownmechanisms (e.g. Stiles, 1967;Cornsweet, 1970) and others onconstraints imposed by the statisticalstructure of the environment (e.g.Gregory, 1968; Marr, 1982; Landy &Movshon, 1991; Wandell, 1995; Geisler& Kersten, 2002; Purves & Lotto,2003).

In previous work, we haveelaborated equivalent illuminantmodels of observer performance fortasks where surface mode or surfacecolor was judged (Speigle & Brainard,1996; Brainard, Brunt, & Speigle,1997; see also Brainard, Wandell, &Chichilnisky, 1993; Maloney & Yang,2001; Boyaci, Maloney, & Hersh, 2003).In such models, the observer isassumed to be correctly performing aconstancy computation, with the oneexception that their estimate of theilluminant deviates from the actualilluminant. The parameterization ofthe observer’s illuminant estimatedetermines the range of performancethat may be explained, with thedetailed calculation then followingfrom an analysis of the physics ofimage formation. Here we present anequivalent illuminant model for howperceived lightness varies with

Bloj et al. 3

surface slant. Our model isessentially identical to thatformulated recently by Boyaci et al.(2003).

Equivalent Illuminant Model

OverviewOur model is derived fromconsideration of an inverse opticscalculation that could achieveconstancy. The inverse opticscalculation begins with parametersthat describe the illuminationgeometry. If these parameters matchthose of the physical scene, thecalculation achieves constancy.Deviations in the model’s parametersfrom those of the scene predictdeviations from constancy. In the nextsections we describe the physicalmodel of illumination and how thismodel can be incorporated into aninverse optics calculation to achieveconstancy. We then show how theformal development leads to aparametric model of observerperformance.

Physical ModelConsider a Lambertian flat matte

standard object1 that is illuminatedby a point2 directional light source.The standard object is oriented at aslant N with respect to a referenceaxis (x-axis in Figure 2). The lightsource is located at a distance dfrom the standard surface. The lightsource azimuth is indicated by D andthe light source declination (withrespect to the z-axis) by D.

The luminance Li, Nof the light

reflected from the standard surface idepends on its surface reflectance ri,its slant N, and the intensity of theincident light E:

Li, N= riE . (1)

When the light arrives only directlyfrom the source, we can write

E = ED (2)

where

ED =

ID sin D[cos( D − N )]

d2 . (3)

Here ID represents the luminousintensity of the light source.Equation 3 applies when−90° ≤ ( D − N ) ≤ 90°. For a purelydirectional source and ( D − N ) outsideof this range, ED = 0.

Figure 2 about here

Figure 2. Reference system centered on the standard object.The standard object is oriented so that its surface normal formsan angle N with respect to the x-axis. The light source islocated at a distance d from this point, the light sourceazimuth (with respect to the x-axis) is D , and the light sourcedeclination (with respect to the z-axis) is D .

In real scenes, light from a sourcearrives both directly and afterreflection off other objects. Forthis reason, the incident light E canbe described more accurately as acompound quantity made of thecontribution of directional light ED

and some diffuse light EA. The termEA provides an approximate descriptionof the light reflected off otherobjects in the scene. We rewriteEquation 2 as

E = ED + EA (4)

and Equation 1 becomes

Li, N

= riID sin D[cos( D − N )]

d2 + EA

. (5)

The luminance of the standard surfaceLi, N

reaches its maximum value when

D − N = 0° and its minimum whenD − N ≥ 90°. In the latter case only

the ambient light EA illuminates thestandard surface.

It is thus useful to simplifyEquation 5 by factoring out amultiplicative scale factor that isindependent of N:

Li, N= ri (cos( D − N ) + FA ) . (6)

In this expression =I D sin D

d 2 and FA

is given by FA =

d2 EA

ID sin D.

Bloj et al. 4

Physical Model FitHow well does the physical model

describe the illumination in ourapparatus? We measured the luminanceof our standard objects under allexperimental slants, and averagedthese over standard objectreflectance. Figure 3 (solid circles)shows the resulting luminances fromeach experiment of the companion paper(Ripamonti et al., 2004) plottedversus the standard object slant. Foreach experiment, the measurements arenormalized to a value of 1 at N = 0°.We denote the normalized luminances by

LN

norm. The solid curves in Figure 3denote the best fit of Equation 6 tothe measurements, where D, FA and were treated as a free parameters andchosen to minimize the mean squarederror between model predictions andmeasured normalized luminances.

Figure 3 about here

Figure 3. The green symbols represent the relative normalizedluminance measured for standard objects used in (Ripamontiet al., 2004) and the colored curves illustrate the fit of themodel described in the text. Top panel corresponds to the lightsource set-up used in Experiments 1 and 2, middle panel toExperiment 3 light source on the left, bottom panel forExperiment 3 light source on the right.

The fitting procedure returns twoestimated parameters of interest: theazimuth D of the light source and theamount FA of ambient illumination.(The scalar simply normalizes thepredictions in accordance with thenormalization of the measurements.)We can represent these parameters in apolar plot, as shown in Figure 4. Theazimuthal position of the plottedpoints represents D, while the radiusv at which the points are plotted isa function of FA:

v =

1FA + 1

. (7)

If the light incident on the standardis entirely directional then theradius of the plotted point will be 1.In the case where the light incidentis entirely ambient, the radius willbe 0.

Figure 4 about here

Figure 4. Light source position estimates of the physical model.Green lines represent the light source azimuth as measured inthe apparatus. In Experiments 1, 2 and 3 -light source on theleft- the actual azimuth was D = -36°. In Experiment 3 -lightsource on the right- the actual azimuth was D = 23°. The redsymbol represents light source azimuth estimated by the modelfor Experiments 1 and 2 ( D = -25°). For the light source onthe left, in Experiment 3, the model estimate is indicated inblue ( D = -30°); for the light source on the right in purple( D = 25°). The radius of the plotted points providesinformation about the relative contributions of directional andambient illumination to the light incident on the standard objectthrough Equation 7. The radius of the outer circle in the plot is1. The parameter values obtained for FA are FA =0.18(Experiments 1 and 2), FA =0.43 (Experiment 3 left), andFA =0.43 (Experiment 3 right).

The physical model provides a goodfit to the dependence of the measuredluminances on standard object slant.It should be noted, however, that therecovered azimuth of the directionallight source differs from our directmeasurement of this azimuth. The mostlikely source of this discrepancy isthat the ambient light arising fromreflections off the chamber walls hassome directional dependence. Thisdependence is absorbed into themodel’s estimate of D.

Equivalent Illuminant ModelSuppose an observer has full

knowledge of the illumination andscene geometry and wishes to estimatethe reflectance of the standardsurface from its luminance. FromEquation 6 we obtain the estimate

˜ r i, N=

Li, N

(cos( D − N )+ FA ). (8)

We use a tilde to denote perceptualanalogs of physical quantities.

To the extent that the physicalmodel accurately predicts theluminance of the reflected light,Equation 8 predicts that theobserver’s estimates of reflectancewill be correct and thus Equation 8predicts lightness constancy. Toelaborate Equation 8 into a parametricmodel that allows failures ofconstancy, we replace the parametersthat describe the illuminant with

Bloj et al. 5

perceptual estimates of theseparameters:

˜ r i, N=

Li, N

(cos( ˜ D − N )+ ˜ F A )

(9)

where ˜ D and ˜ F A are perceptual analogs

of D and FA. Note that thedependence of ˜ r i, N

on slant inEquation 9 is independent of ri.

Equation 9 predicts an observer’sreflectance estimates as a function ofsurface slant, given the parameters ˜

D

and ˜ F A of the observer’s equivalentilluminant. These parameters describethe illuminant configuration that theobserver uses in his or her inverseoptics computation.

Our data analysis procedureaggregates observer matches overstandard object reflectance to producerelative normalized matches r

N

norm.The relative normalized matchesdescribe the overall dependence ofobserver matches on slant. To linkEquation 8 with the data, we assumethat the normalized relative matchesobtained in our experiment (seeAppendix of Ripamonti et al., 2004)are proportional to the computed ˜ r i, N

,leading to the model prediction

r N

norm =L

N

norm

(cos( ˜ D − N )+ ˜ F A )

(10)

where is a constant ofproportionality that is determined aspart of the model fitting procedure.In Equation 10 we have substituted

LN

norm for Li, N since the contribution of

surface reflectance ri can be absorbedinto . (11)

Equation 10 provides a parametricdescription of how our measurements ofperceived lightness should depend onslant. By fitting the model to themeasured data, we can evaluate howwell the model is able to describeperformance, and whether it cancapture the individual differences weobserve. In fitting the model, thetwo parameters of interest are ˜

D and˜ F A, while the parameter simply

accounts for the normalization of thedata.

In generating the modelpredictions, values for N and L

N

norm

are taken as veridical physicalvalues. It would be possible todevelop a model where these were alsotreated as perceptual quantities andthus fit to the data. Withoutconstraints on how ˜

N and ˜ L N

norm arerelated to their physicalcounterparts, however, allowing theseas parameters would lead to excessivedegrees of freedom in the model. Inour slant matching experiment,observer’s perception of slant wasclose to veridical and thus using thephysical values of N seems justified.We do not have independentmeasurements of how the visual systemregisters luminance.

Model Fit

Fitting the modelFor each observer, we used

numerical search to fit the model tothe data. The search procedure foundthe equivalent illuminant parameters˜

D (light source azimuth) and ˜ F A(relative ambient) as well as theoverall scaling parameter thatprovided the best fit to the data.The best fit was determined asfollows. For each of the threesessions k = 1,2,3 we found thenormalized relative matches for thatsession, r k, N

norm. We then found theparameters that minimized the meansquared error between the model’sprediction and these r k, N

norm. Thereason for computing the individualsession matches and fitting to these,rather than fitting directly to theaggregate r

N

norm is that the formerprocedure allows us to compare themodel’s fit to that obtained byfitting the session data at each slantto its own mean.

Bloj et al. 6

Model FitModel fit results are illustrated

in the left hand columns of Figures 5to 10. The dot symbols (green forNeutral instructions and purple forPaint Instructions) are observers’normalized relative matches and theorange curve in each panel shows thebest fit of our model. We also showthe predictions for luminance andconstancy matches as, respectively, ablue or red dashed line. The righthand columns of Figures 5 to 10 showthe model’s ˜

D and ˜ F A for eachobserver, using the same polar formatintroduced in Figure 4.

Figure 5 about here

Figure 5. Model fit to observers’ relative normalized matches.In the left column the green dots represent observers’ relativenormalized matches as a function of slant for Experiment 1.Error bars indicate 90% confidence intervals. The orange curveis the model’s best fit for that observer. The blue dashed curverepresents predictions for luminance matches and the reddashed line for constancy matches. The right column showsthe equivalent illuminant parameters (green symbols) in thesame polar format introduced in Figure 4. The polar plot alsoshows the illuminant parameters obtained by fitting thephysical model to the measured luminances (red symbols).The numbers at the top left of each data plot are the error-based constancy index for the observer, while those at the topleft of the polar plots are the corresponding model-basedindex, derived from the equivalent illuminant parameters.

Figure 6 about here

Figure 6. Model fit to observers’ relative normalized matchesfor Experiment 2. Same format as Figure 5.

Figure 7 about here

Figure 7. Model fit to observers’ relative normalized matchesfor Experiment 3 (light on the left, Neutral instructions). Sameformat as Figure 5.

Figure 8 about here

Figure 8. Model fit to observers’ relative normalized matchesfor Experiment 3 (light on the right, Neutral instructions). Sameformat as Figure 5.

Figure 9 about here

Figure 9. Model fit to observers’ relative normalized matchesfor Experiment 3 (light on the left, Paint instructions). Sameformat as Figure 5.

Figure 10 about here

Figure 10. Model fit to observers’ relative normalized matchesfor Experiment 3 (light on the right, Paint instructions). Sameformat as Figure 5.

With only a few exceptions, theequivalent illuminant model capturesthe wide range of performanceexhibited by individual observers inour experiment. To evaluate thequality of the fit, we can compare themean squared error for the equivalentilluminant equiv

2 model to thevariability in the data. To make thiscomparison, we also fit the r k, N

norm ateach session and slant by their ownmeans. For each observer, theresulting mean squared error prec

2 is alower bound on the mean squared errorthat could be obtained by any model.A figure of merit for the equivalentilluminant model is then quantity

prec = equiv

prec

. This quantity should be

near unity if the model fits well, andvalues greater than unity indicate fiterror in units yoked to the precisionof the data. Across all our observersand light source positions, the meanvalue of equiv was 1.23, indicating agood but not perfect fit.

For comparison, we also computed values associated with four othermodels. These are:

a) luminance matching: r N

norm = LN

norm

b) lightness constancy: r N

norm =

c) mixture: r N

norm = LN

norm + (1− )( )d) quadratic: r

N

norm = N2 + N + .

The mixture model describes observerswhose responses are an additivemixture of luminance matching andlightness constancy matches. If thismodel fit well, the mixing parameter could be interpreted as describing

the matching strategy adopted bydifferent observers. The quadraticmodel has no particular theoreticalsignificance, but has the same numberof parameters as our equivalentilluminant model and predicts smoothlyvarying functions of N. The solidbars in Figure 11 show the mean

Bloj et al. 7

values for all five models. We seethat the error for the equivalentilluminant model is lower than thatfor the four comparison models. Thisdifference is statisticallysignificant at the p < 0.0001 for allmodels, as determined by sign test onthe values obtained for eachobserver/light source positioncombination.


Figure 11. Evaluation of model fits. Solid bars show the mean values obtained when the matching data for each slant,

session and observer are fitted by: the equivalent illuminantmodel and the four comparison models described in the text.

Also shown is the value when each r N

norm is fit by its own

mean. This value is labeled Precision and is constrained bythe definition of to be unity. No model can have an lessthan unity. The hatched bars show the cross-validation

values.

The various models evaluated abovehave different numbers of parameters.For this reason, it is worth askingwhether the equivalent illuminantmodel performs better simply becauseit overfits the data. Answering thisquestion is difficult. Selectionamongst non-nested and/or non-linearmodels remains a topic of activeinvestigation (see special issue onmodel selection, J. Math. Psych, vol.44, issue 1, 2000) and the literaturedoes not yet provide a recipe. Herewe adopt a cross-validation approach.

Our measurements consist of the

r k, N

norm measured in three sessions. Weaveraged the data from each possiblepair of two sessions and used theresult to fit each model. Then foreach model and session pair weevaluated how well the model fit thesession data that had been excludedfrom the fitting procedure, using thesame metric described above. Theintuition is that a model thatoverfits the data should generalizepoorly and have high cross-validation values, while a model that captures

structure in the data shouldgeneralize well and have low cross-validation values.

The hatched bars in Figure 11 showthe cross-validation values weobtained. The equivalent illuminant

model continues to perform best. Notethat the cross-validation valueobtained when the data for eachsession is predicted from the mean ofthe other two sessions (labeledPrecision) is higher than thatobtained for the equivalent illuminantmodel. This difference isstatistically significant (sign test,p < 0.005).

Although the equivalent illuminantmodel provides the best fit amongthose we examined, it does not accountfor all of the systematic structure inthe data. ANOVAs conducted on themodel residuals indicated that thesedepend on surface slant in astatistically significant manner forseveral of our conditions (Exp. 1,p = 0.14; Exp. 2, p = 0.14; Exp. 3Left Neutral, p < 0.005; Exp. 3 RightNeutral, p < 0.005; Exp. 3 Left Paint,p < 0.1; Exp. 3 Right Paint,p < 0.005). The systematic nature ofthe residuals was more salient for allfour of the comparison models(p < 0.001 for all models/conditions)than for the equivalent illuminantmodel.

Discussion

Using the ModelThe equivalent illuminant allows

interpretation of the large individualdifferences observed in ourexperiments. In the context of themodel, these differences are revealedas variation in the equivalentilluminant model parameters ˜

D and ˜ F A,rather than as a qualitativedifference in the manner in whichobservers perform the matching task.In the polar plots we see that foreach condition, the equivalentilluminant model parameters lieroughly between the origin and thecorresponding physical illuminantparameters. Observers whose dataresembles luminance matching haveparameters that plot close to theorigin, while those whose dataresemble constancy matching haveparameters that plot close to those ofthe physical illuminant. This pattern

Bloj et al. 8

in the data reflects the fact thatobservers’ performance lies betweenthat of luminance matching andlightness constancy. The fact thatmany observers have illuminantparameters that differ from thecorresponding physical values could beinterpreted as an indication of thecomputational difficulty of estimatinglight source position and relativeambient from image data.

Various patterns in the raw datashown by many observers, particularlythe sharp drop in match for N = 60°when the light is on the left and thenon-monotonic nature of the matcheswith increasing slant, require nospecial explanation in the context ofthe equivalent illuminant model. Bothof these patterns are predicted by themodel for reasonable values of theparameters. Indeed, striking to uswas the richness of the model’spredictions for relatively smallchanges in parameter values.

A question of interest inExperiment 3 was whether observers aresensitive to the actual position ofthe light source. Comparison of ˜

D

across changes in the light sourceposition indicates that they are. Theaverage value of ˜

D when the lightsource was on the left in Experiment 3was -35°, compared to 16° when it wason the right. The shift in equivalentilluminant azimuth of 51° iscomparable to the corresponding shiftin the physical model parameter (55°).

Model-Based Constancy IndexIn the companion paper, we

developed a constancy index based oncomparing the fit error for luminancematching and constancy. Such indicesprovide a summary of what the dataimply about lightness constancy. Atthe same time, any given constancyindex is of necessity somewhatarbitrary. It is therefore ofinterest to derive a model-basedconstancy index and compare it withthe error-based index.

Let the vector

v =vsin D

vcos D

(12)

be a function of the physical model’sparameters D and FA, with the scalarv computed from FA using Equation 7above. Let the vector ˜ v be theanalogous vector computed from theequivalent illuminant model parameters˜

D and ˜ F A. Then we define the modelbased constancy index as

CIm = 1 -

v − ˜ v

v . (13)

This index takes on a value of 1 whenthe equivalent illuminant modelparameters match the physical modelparameters and a value near 0 when theequivalent illuminant model parameter˜ F A is very large. This latter casecorresponds to where the modelpredicts luminance matching.

We have computed this CIm for eachobserver/condition, and the resultingvalues are indicated on the top leftof each polar plot in Figures 5-10.The model based constancy index rangesfrom 0.23 to 0.91, with a mean of 0.57a median of 0.57. These values arelarger than those obtained with theerror based index (mean/median 0.40).Figure 12 shows a scatter plot of thetwo indices, which are correlated at r= 0.73. The discrepancy between thetwo indices provides a sense of theprecision with which they should beinterpreted. Given the computationaldifficulty of recovering lightinggeometry from images, we regard theaverage degree of constancy shown bythe observers (~0.40 – ~0.57)as afairly impressive achievement. Thelarge individual variability inperformance remains clear in Figure12.


Figure 12. Scatter plot of error-based versus model-basedconstancy indices. Each point represents the two indices ofone observer. For Experiment 3, indices for left and right lightsource positions are plotted separately.

Bloj et al. 9

Interpreting the Model ParametersThe equivalent illuminant model has

two parameters, ˜ D and ˜ F A, that

describe the lighting geometry. Theseparameters are not, however, set bymeasurements of the physical lightinggeometry but are fit to eachobserver’s data. Given the equivalentilluminant parameters, the modelpredicts the lightness matches throughan inverse optics calculation.

It is tempting to associate theparameters ˜

D and ˜ F A with observers’consciously accessible estimates ofthe illumination geometry. Since ourexperiments do not explicitly measurethis aspect of perception, we have noempirical basis for making theassociation. In interpreting theparameters as observer estimates ofthe illuminant, it is important tobear in mind that they are derivedfrom surface lightness matching dataand thus, at present, should betreated as illuminant estimates onlyin the context of our model of surfacelightness. It is possible that afuture explicit comparison couldtighten the link between the derivedparameters and conscious perception ofthe illuminant. Prior attempts tomake such links between implicit andexplicit illumination perception,however, have not led to positiveresults (see e.g. Rutherford &Brainard, 2002).

Independent of the connectionbetween model parameters andexplicitly judged illuminationproperties, equivalent illuminantmodels are valuable to the extent a)that the provide a parsimoniousaccount of rich data sets and b) thattheir parameters can be predicted bycomputational algorithms that estimateilluminant properties (e.g. Brainard,Kraft, & Longère, 2003; Brainard etal., 2004). As computationalalgorithms for estimating illuminationgeometry become available, our hope isthat these may be used in conjunctionwith the type equivalent illuminantmodel presented here to predictperceived surface lightness directlyfrom the image data.

AcknowledgementsThis work was supported NIH EY

10016. We thank B. Backus, H. Boyaci,L. Maloney, R. Murray, J. Nachmias,and S. Sternberg for helpfuldiscussions.

References

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Ripamonti, C., Bloj, M., Mitha, K., Greenwald, S.,Hauck, R., Maloney, S. I., & Brainard, D. H.(2004). Measurements of the effect of surfaceslant on perceived lightness. Journal of Vision,submitted.

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Notes 1 A Lambertian surface is a uniformlydiffusing surface with constantluminance regardless of the direction

from which is viewed (Wyszecki andStiles, 1982).2A light source whose distance fromthe illuminated object is at least 5times its main dimension is consideredto be a good approximation of a pointlight source (Kaufman & Christensen,1972).

Bloj et al., Figure 1

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-60 -30 0 30 60

0.4

0.8

1.2

1.60.33

-60 -30 0 30 60

0.4

0.8

1.2

1.60.25

-60 -30 0 30 60

0.4

0.8

1.2

1.60.17

-60 -30 0 30 60

0.4

0.8

1.2

1.60.39

-60 -30 0 30 60

0.4

0.8

1.2

1.60.29

-60 -30 0 30 60

0.4

0.8

1.2

1.60.54

-60 -30 0 30 60

0.4

0.8

1.2

1.60.49

Rel

ativ

e m

atch

refle

ctan

ce

Standard object slant


0

-90 90

-60 60

-30 30

LEF0.32

0

-90 90

-60 60

-30 30

IBO0.49

0

-90 90

-60 60

-30 30

HWC0.59

0

-90 90

-60 60

-30 30

JPL0.62

0

-90 90

-60 60

-30 30

KIR0.75

0

-90 90

-60 60

-30 30

NMR0.77

0

-90 90

-60 60

-30 30

MRG0.91

-60 -30 0 30 60

0.4

0.8

1.2

1.60.36

-60 -30 0 30 60

0.4

0.8

1.2

1.60.49

-60 -30 0 30 60

0.4

0.8

1.2

1.60.52

-60 -30 0 30 60

0.4

0.8

1.2

1.60.24

-60 -30 0 30 60

0.4

0.8

1.2

1.60.62

-60 -30 0 30 60

0.4

0.8

1.2

1.60.54

-60 -30 0 30 60

0.4

0.8

1.2

1.60.63

Rel

ativ

e m

atch

refle

ctan

ce



0

-90 90

-60 60

-30 30

FGP0.23

0

-90 90

-60 60

-30 30

EKS0.42

0

-90 90

-60 60

-30 30

DDB0.41

0

-90 90

-60 60

-30 30

ALR0.51

0

-90 90

-60 60

-30 30

BMZ0.52

0

-90 90

-60 60

-30 30

GPW0.55

0

-90 90

-60 60

-30 30

CPK0.64

-60 -30 0 30 60

0.4

0.8

1.2

1.60.41

-60 -30 0 30 60

0.4

0.8

1.2

1.60.38

-60 -30 0 30 60

0.4

0.8

1.2

1.60.29

-60 -30 0 30 60

0.4

0.8

1.2

1.60.45

-60 -30 0 30 60

0.4

0.8

1.2

1.60.21

-60 -30 0 30 60

0.4

0.8

1.2

1.60.3

-60 -30 0 30 60

0.4

0.8

1.2

1.60.41

Rel

ativ

e m

atch

refle

ctan

ce



0

-90 90

-60 60

-30 30

FGP0.68

0

-90 90

-60 60

-30 30

EKS0.80

0

-90 90

-60 60

-30 30

DDB0.58

0

-90 90

-60 60

-30 30

ALR0.72

0

-90 90

-60 60

-30 30

BMZ0.50

0

-90 90

-60 60

-30 30

GPW0.82

0

-90 90

-60 60

-30 30

CPK0.67

-60 -30 0 30 60

0.4

0.8

1.2

1.60.42

-60 -30 0 30 60

0.4

0.8

1.2

1.60.39

-60 -30 0 30 60

0.4

0.8

1.2

1.60.44

-60 -30 0 30 60

0.4

0.8

1.2

1.60.54

-60 -30 0 30 60

0.4

0.8

1.2

1.60.31

-60 -30 0 30 60

0.4

0.8

1.2

1.60.37

-60 -30 0 30 60

0.4

0.8

1.2

1.60.53

Rel

ativ

e m

atch

refle

ctan

ce



0

-90 90

-60 60

-30 30

HWK0.31

0

-90 90

-60 60

-30 30

JHO0.30

0

-90 90

-60 60

-30 30

IQB0.33

0

-90 90

-60 60

-30 30

LPS0.41

0

-90 90

-60 60

-30 30

KVA0.49

0

-90 90

-60 60

-30 30

MOG0.56

0

-90 90

-60 60

-30 30

NPY0.57

-60 -30 0 30 60

0.4

0.8

1.2

1.60.19

-60 -30 0 30 60

0.4

0.8

1.2

1.60.26

-60 -30 0 30 60

0.4

0.8

1.2

1.60.35

-60 -30 0 30 60

0.4

0.8

1.2

1.60.33

-60 -30 0 30 60

0.4

0.8

1.2

1.60.51

-60 -30 0 30 60

0.4

0.8

1.2

1.60.28

-60 -30 0 30 60

0.4

0.8

1.2

1.60.51

Rel

ativ

e m

atch

refle

ctan

ce



0

-90 90

-60 60

-30 30

HWK0.53

0

-90 90

-60 60

-30 30

JHO0.56

0

-90 90

-60 60

-30 30

IQB0.66

0

-90 90

-60 60

-30 30

LPS0.70

0

-90 90

-60 60

-30 30

KVA0.74

0

-90 90

-60 60

-30 30

MOG0.76

0

-90 90

-60 60

-30 30

NPY0.71

-60 -30 0 30 60

0.4

0.8

1.2

1.60.41

-60 -30 0 30 60

0.4

0.8

1.2

1.60.36

-60 -30 0 30 60

0.4

0.8

1.2

1.60.37

-60 -30 0 30 60

0.4

0.8

1.2

1.60.45

-60 -30 0 30 60

0.4

0.8

1.2

1.60.5

-60 -30 0 30 60

0.4

0.8

1.2

1.60.48

-60 -30 0 30 60

0.4

0.8

1.2

1.60.61

Rel

ativ

e m

atch

refle

ctan

ce



0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Precision Equiv Quad Mixture Lum Const

Mea

n F

it E

rror

Model

Figure 11

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

Err

or C

Is

Model CIs

Figure 12

an equivalent illuminant model for the effect of surface...

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