David H. Brainard Department of Psychology University of ... · David H. Brainard Department of...

Color Constancy

David H. BrainardDepartment of PsychologyUniversity of Pennsylvania

3815 Walnut StreetPhiladelphia, PA 19104

[email protected]

Chapter to appear in The Visual Neurosciences.

D R A F T: May 2002

Brainard Color Constancy

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Introductions

Color is used to recognize and describe objects. When givingdirections, we might provide the detail that the destination is a yellowhouse. When judging the ripeness of a fruit, we might evaluate itscolor. The ability to perceive objects as having a well-defined color isquite remarkable. To understand why, it is necessary to consider howinformation about object spectral properties is represented in theretinal image.

A scene is a set of illuminated objects. In general the illuminationhas a complex spatial distribution, so that the illuminant falling on oneobject in the scene may differ from that falling on another. None-the-less, a useful point of departure is to consider the case where theillumination is uniform across the scene, so that it may becharacterized by its spectral power distribution, E( )l . This functionspecifies how much power the illuminant contains at each wavelength.The illuminant reflects off objects to the eye, where it is collected andfocused to form the retinal image. It is the image that is explicitlyavailable for determining the composition of the scene.

Object surfaces differ in how they absorb and reflect light. Ingeneral reflection depends on wavelength, the angle of the incidentlight (relative to the surface normal), and the angle of the reflectedlight (Foley et al. 1990). It is again useful to simplify and neglectgeometric considerations, so that each object surface is characterizedby a spectral reflectance function, S( )l . This function specifies whatfraction of incident illumination is reflected from the object at eachwavelength.

The light reflected to the eye from each visible scene location iscalled the color signal. For the simplified imaging model describedabove, the spectral power distribution of the color signal C( )l is readilycalculated from the illuminant spectral power distribution and surfacereflectance function:

C E S( ) ( ) ( )l l l= . (1)

The retinal image consists of the color signal incident at each location,after blurring by the eye’s optics. In the treatment here, such blurringmay be safely ignored.

The imaging model expressed by Eqn. 1 assumes that the lightsource is spatially uniform, that the objects are flat and coplanar, andthe surface reflectances are Lambertian. It is sometimes referred toas the Mondrian World imaging model. The assumptions of theMondrian World never hold for real scenes. A more realistic


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formulation would include description of the spatial distribution of theilluminant, the geometry of the scene, and how each object’s spectralreflectance depends on the direction of the incident and reflected light(Foley et al., 1990; also Figure 2 below). None-the-less, the MondrianWorld is rich enough to provide a useful framework for initial analysis.

The form of Eqn. 1 makes explicit the fact that two distinct physicalfactors, the illuminant and the surface reflectance, contribute in asymmetric way to the color signal. One of these factors, the surfacereflectance, is intrinsic to the object and carries information about itsidentity and properties. The other factor, the illuminant, is extrinsic tothe object and provides no information about the object.

Given that the color signal at an image location confoundsilluminant and surface properties, how is it possible to perceive objectsas having a well-defined color? Indeed, the form of Eqn. 1 suggeststhat changes in illuminant can masquerade perfectly as changes inobject surface reflectance, so that across conditions where theilluminant varies one might expect large changes in the appearance ofa fixed object. This physical process is illustrated by Figure 1. Each ofthe two patches shown at the top of the figure corresponds to a regionof a single object, imaged under a different outdoor illuminant. Whenthe two patches are seen in isolation, their color appearance is quitedifferent: there is enough variation in natural daylight that the colorsignal is substantially ambiguous about object surface properties.

When the two patches are seen in the context of the images fromwhich they were taken, the variation in color appearance is reduced.This stabilization of appearance is by no means complete in the figure,where the reader views small printed images that are themselves partof a larger illuminated environment. For an observer standing in frontof the home shown, however, the variation in perceived color isminimal and not normally noticed. This suggests that the visualsystem attempts to resolve the ambiguity inherent in the color signalby analyzing many image regions jointly: the full image context isused to produce a stable perceptual representation of object surfacecolor. This ability is referred to as color constancy.


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Figure 1. Same objects imaged under two naturalilluminants. Top) The patches show a rectangular regionextracted from images of the same object under differentoutdoor illuminants. Bottom) The images from which thepatches were taken. Images were acquired by the author inMerion Station, PA using a Nikon CoolPix 995 digital camera.The automatic white balancing calculation that is a normal partof the camera’s operation was disabled during imageacquisition.

This chapter is about human color constancy. The literature oncolor constancy is vast, extending back at least to the eighteenthcentury (see Mollon, in press), and this chapter does not attempt asystematic review. Rather, the goal is to provide an overview of howhuman color constancy can be studied experimentally and how it canbe understood. The next section presents an extended example ofhow constancy is measured in the laboratory. The measurementsshow both circumstances where constancy is good and those where itis not; a characterization of human color vision as either“approximately color constant” or “not very color constant” is toosimple. Rather, we must characterize when constancy will be goodand when it will not. The discussion outlines two current approaches.

Measuring Constancy

This section illustrates how constancy may be measured bydescribing experiments conducted by Kraft & Brainard (1999; see alsoBrainard 1998; Kraft, Maloney, & Brainard 2002). Before treating thespecific experimental design, however, some general remarks are inorder.

Several distinct physical processes can cause the illuminationimpinging on a surface to vary. The images in Figure 1 above


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illustrate one such process. The images were taken at different times,and the spectra of the illuminant sources change. Color constancyacross illumination changes that occur over time is called successivecolor constancy.

Geometric factors can also cause the illumination impinging on asurface to change. This is illustrated by Figure 2. All of the effectsshown occur without any change in the spectral of the light sourcesbut instead are induced by the geometry of the light sources andobjects. Color constancy across illumination changes that occur withina single scene is called simultaneous color constancy.

The visual system’s ability to achieve simultaneous constancy neednot be easily related to its ability to achieve successive constancy.Indeed, fundamental to simultaneous constancy is some sort ofsegmentation of the image into regions of common illumination, whilesuch segmentation is not obviously necessary for successive constancy(see Adelson, 1999). Often results and experiments about successiveand simultaneous constancy are compared and contrasted withoutexplicit acknowledgment the two may be quite different; keeping thedistinction in mind as one considers constancy can reduce confusion.This chapter will focus on successive constancy, as many of the keyconceptual issues can be introduced without the extra richness ofsimultaneous constancy. The discussion returns briefly tosimultaneous constancy.

In the introduction, constancy was cast in terms of the stability ofobject color appearance, and this is the sense in which theexperiments presented below assess it. Some authors (Brainard &Wandell, 1988; D’Zmura & Mangalick, 1994; Foster & Nascimento,1994; Khang & Zaidi, 2002) have suggested that constancy might bestudied through performance (e.g. object identification) rather thanthrough appearance per se. One might expect appearance to play animportant role in identification, but reasoning might also be involved.Although the study of constancy using performance-based methods isan interesting direction, this chapter is restricted to measurements andtheories of appearance.


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Figure 2. Image formation. Each set of square patchesaround the side of the image illustrate variation in the lightreflected to the eye when surface reflectance is held fixed.Gradient: The two patches shown were extracted from theupper left (L) and lower right (R; above table) of the back wallof the scene. Shadow. The two patches were extracted fromthe tabletop in direct illumination (D) and shadow (S). Shape:The three patches shown were extracted from two regions of thesphere (T and B; center top and right bottom respectively) andfrom the colored panel directly above the sphere (P; the panel isthe leftmost of the four in the bottom row). Both the sphereand panel have the same simulated surface reflectance function.Pose and Indirect Illum. The four patches were extracted fromthe three visible sides of the cube (R, L, and T; right, left andtop visible sides respectively) and from the left side of thefolded paper located between the cube and sphere (I). Thesimulated surface reflectance of all sides of the cube and of theleft side of the folded paper are identical. The image wasrendered from a synthetic scene description using the Radiancecomputer graphics package (Larson & Shakespeare, 1998).There were two sources of illumination in the simulated scene: adiffuse illumination that would appear bluish if viewed inisolation and a directional illumination (from upper left) thatwould appear yellowish if viewed in isolation. All of the effectsillustrated by this rendering are easily observed in naturalscenes.

R L T I

Pose and Indirect Illum. Shape

PT B

Shadow

D S

Gradient

L R


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An example experiment

Figure 3 shows the basic experimental setup used by Kraft &Brainard (1999). Subjects viewed a collection of objects contained inan experimental chamber. The chamber illumination was provided bytheater lamps. The light from the lamps passed through a diffuserbefore entering the chamber, so that the overall effect was of a singlediffuse illuminant. Each lamp had either a red, green, or blue filter,and by varying the intensities of the individual lamps the spectrum ofthe chamber illumination could be varied. Because the light in thechamber was diffuse, the viewing environment provided a roughapproximation to Mondrian World conditions.

The far wall of the experimental chamber contained a test patch.Physically, this was a surface of low neutral reflectance so that undertypical viewing conditions it would have appeared dark gray. The testpatch was illuminated by the ambient chamber illumination and also bya separate projector. The projector beam was precisely aligned withthe edges of the test patch and consisted of a mixture of red, green,and blue primaries. By varying the amount of each primary in themixture, the light reflected to the observer from the test patch couldbe varied independently of the rest of the image. The effect ofchanging the projected light was to change the color appearance of thetest, as if it had been repainted. The apparatus thus functioned tocontrol the effect described by Gelb (1950; see also Koffka, 1935;Katz, 1935), wherein using a hidden light source to illuminate a paperdramatically changes its color appearance.

The observer’s task in Kraft & Brainard’s (1999) experiments was toadjust the test patch until it appeared achromatic. Achromaticjudgments have been used extensively in the study of colorappearance (e.g. Helson & Michels, 1948; Werner & Walraven, 1982;Chichilnisky & Wandell, 1996). During the adjustment, the observercontrolled the chromaticity of the test patch while its luminance washeld constant. In essence, the observer chose the test patchchromaticity that appeared gray, when seen in the context set by therest of the experimental chamber. Whether the test patch appearedlight gray or dark gray depended its luminance. This was heldconstant during individual adjustments but varied betweenadjustments. For conditions where the luminance of the test patch islow relative to its surroundings, Brainard (1998) found no dependenceof the chromaticity of the achromatic adjustment as a function of testpatch luminance. This independence does not hold when more


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luminous test patches are used (Werner & Walraven, 1982;Chichilnisky & Wandell, 1996).

R

G B

R

G B

Test patch

Projectioncolorimeter

Figure 3. Schematic diagram of the experimentalapparatus used in the experiments of Kraft and Brainard.An experimental chamber was illuminated by computercontrolled theater lamps. Different filters were placed overindividual lamps, so that by varying their relative intensity theoverall spectral power distribution of the chamber illuminationcould be varied. The light from the lamps was passed through adiffuser, producing a fairly homogenous illumination. Theobserver viewed a test patch on the far wall of the chamber.The test patch was illuminated by the ambient chamberillumination and also by a beam from a projection colorimeter.The beam from the colorimeter was not explicitly visible, so thatthe perceptual effect of varying it was to change the apparentsurface color of the test patch.


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The data from the experiment are conveniently represented usingthe standard 1931 CIE chromaticity diagram. Technical explanationsof this diagram and its basis in visual performance are widely available(e.g. CIE, 1986; Kaiser & Boynton, 1996; Brainard, 1995), but its keyaspects are easily summarized. Human vision is trichromatic, so thata light C( )l may be matched by a mixture of three fixed primaries:

C X P Y P Z P( ) ~ ( ) ( ) ( )l l l l1 2 3+ + . (2)

In this equation P1( )l , P2( )l , and P3( )l are the spectra of the threeprimary lights being mixed and the scalars X , Y , and Z specify theamount of each primary in the mixture. The symbol ~ indicates visualequivalence. When we are concerned with human vision,standardizing a choice of primary spectra allows us to specify aspectrum compactly by its tristimulus coordinates X , Y , and Z . TheCIE chromaticity diagram is based on a set of known primariestogether with a standard of performance that allows computation ofthe tristimulus coordinates of any light from its spectrum. Thechromaticity diagram, however, represents lights with only twocoordinates x and y. These chromaticity coordinates are simplynormalized versions of the tristimulus coordinates:

xX

X Y Zy

Y

X Y Z=

+ +=

+ +, . (3)

The normalization removes from the representation all informationabout the overall intensity of the spectrum while preserving theinformation about the relative spectrum that is relevant for humanvision.

Figure 4 shows data from two experimental conditions. Eachcondition is defined by the scene within which the test patch wasadjusted. The two scenes, labeled Scene 1 and Scene 2 are shown atthe top of the figure. The scenes were sparse but had visible three-dimensional structure. The surface lining the chamber was the samein the two scenes, but the spectrum of the illuminant differed. Thedata plotted for each condition are the chromaticity of the illuminant(open circles) and the chromaticity of the observers’ achromaticadjustments (closed circles).

The points plotted for the illuminant are the chromaticity of theilluminant as measured at the test patch location, when the projectioncolorimeter was turned off. These represent the chromaticity of theambient illumination in the chamber, which was approximatelyuniform. The fact that the illuminant was changed across the twoscenes is revealed in the figure by the shift between the open circles.


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The plotted achromatic points are the chromaticity of the lightreflected to the observer when the test appeared achromatic. Thislight was physically constructed as the superposition of reflectedambient light and reflected light from the projection colorimeter.Across the two scenes, the chromaticity of the achromatic point shiftsin a manner roughly commensurate with the shift in illuminantchromaticity.

Relation of the data to constancy

What do the data plotted in Figure 4 say about color constancyacross the change from Scene 1 to Scene 2? A natural but misleadingintuition is that the large shift in the achromatic locus shown in thefigure reveals a large failure of constancy. This would be true if thedata plotted represented directly the physical properties of the surfacethat appears achromatic. As noted above, however, the data plotteddescribe the spectrum of the light reaching the observer. To relate thedata to constancy, it is necessary to combine information from themeasured achromatic points and the illuminant chromaticities.

Suppose that the observer perceives the test patch as a surfaceilluminated with the same ambient illumination as the rest of thechamber. Introspection and some experimental evidence support thisassumption (Brainard, Brunt, & Speigle, 1997). The data from Scene 1can then be used to infer the spectral reflectance of an equivalentsurface. The equivalent surface would have appeared achromatic hadit been placed at the test patch location with the projection colorimeterturned off.

Let the reflectance function of the equivalent surface be ˜( )S l . Thisfunction must be such that the chromaticity of E S1( ) ˜( )l l is the same asthe chromaticity of the measured achromatic point, where E1( )l is theknown spectrum of the ambient illuminant in Scene 1. It isstraightforward to find functions ˜( )S l that satisfy this constraint. Theinset to Figure 4 shows one such. The function ˜( )S l is referred to asthe equivalent surface reflectance corresponding to the measuredachromatic point.


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0.2

0.3

0.4

0.5

0.2 0.3 0.4 0.5

IlluminantAchromaticConstancy

CIE

y c

hrom

atic

ity

CIE x chromaticity

400 450 500 550 600 650 7000

0.2

0.4

Ref

lect

ance

Wavelength (nm)

Scene 1 Scene 2

Figure 4. Basic data from achromatic adjustmentexperiment. The images at the top of the figure show theobserver’s view of two scenes, labeled 1 and 2. The test patchis visible in each image. The projection colorimeter was turnedoff at the time the images were acquired, so the images do notshow the results of observers’ achromatic adjustments. Thechromaticity diagram shows the data from achromaticadjustments of the test patch made in the context of the twoscenes. The open circles show the chromaticity of the illuminantfor each scene. The illuminant for Scene 1 plots to the lower leftof the illuminant for Scene 2. The closed circles show thechromaticity of the mean achromatic adjustments of fourobservers. Where visible, the error bars indicate +/- 1 standarderror. The surface reflectance function plotted in the inset atthe right of the figure shows the equivalent surface reflectance˜( )S l computed from the data obtained in Scene 1. The closed

diamond shows the color constant prediction for the achromaticadjustment in Scene 2, given the data obtained for Scene 1.See explanation in the text.


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The equivalent surface reflectance ˜( )S l allows us to predict theperformance of a color constant observer for other scenes. To aconstant observer, any given surface should appear the same whenembedded in any scene. More specifically, a surface that appearsachromatic in one scene should remain so in others. Given the datafor Scene 1, the chromaticity of the achromatic point for a test patch inScene 2 should be the chromaticity of E S2( ) ˜( )l l , where E2( )l is thespectrum of the illuminant in Scene 2. This prediction is shown inFigure 4 by the closed diamond.

Although the measured achromatic point for Scene 2 does notagree precisely with the constancy prediction, the deviation is smallcompared to the deviation that would be measured for an observerwho had no constancy whatsoever. For such an observer, theachromatic point would be invariant across changes of scene. Thusthe data shown in Figure 4 indicate that observers are approximatelycolor constant across the two scenes studied in the experiment.

Brainard (1998) developed a constancy index that quantifies thedegree of constancy revealed by data of the sort presented in Figure 4.The index takes on a value of 0 for no adjustment and 1 for perfectconstancy, with intermediate values for intermediate performance.For the data shown in 4, the constancy index is 0.83. This high valueseems consistent with our everyday experience that the colors ofobjects remain stable over changes of illuminant, but that the stabilityis not perfect.

A paradox and its resolution

The introduction argued that illuminant and surface information isperfectly confounded in the retinal image. The data shown in Figure 4indicate that human vision can separate these confounded physicalfactors and achieve approximate color constancy. This presents aparadox. If the information is perfectly confounded, constancy isimpossible. If constancy is impossible, how can the visual system beachieving it?

The resolution to this paradox is found by considering restrictionson the set of scenes over which constancy holds. Figure 5 shows aschematic diagram of the set of all scenes, represented by the thickouter boundary. Each point within this boundary represents a possiblescene, that is a particular choice of illuminant and surface reflectances.The closed circles represent the two scenes used in the experimentdescribed above. These are labeled Scene 1 and Scene 2 in the figure.


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Denote the retinal image produced from Scene 1 as Image 1. Manyother scenes could have produced this same image. This subset ofscenes is indicated in the figure by the hatched ellipse that enclosesScene 1. This ellipse is labeled Image 1 in the figure. It also containsScene 1̃, indicated by an open circle in the figure. Scenes 1 and 1̃produce the same image and cannot be distinguished by the visualsystem.

Similarly, there is a separate subset of scenes that produce thesame image (denoted Image 2) as Scene 2. This subset is alsoindicated by a hatched ellipse. A particular scene consistent withImage 2 is indicated by the open circle labeled Scene 2̃. As withScenes 1 and 1̃, Scenes 2 and 2̃ cannot be distinguished from eachother by the visual system.

The open ellipse enclosing each solid circle shows a different subsetof scenes to which it belongs. These are scenes that share a commonilluminant. The open ellipse enclosing Scene 1 indicates all scenesilluminated by E1( )l , while the open ellipse enclosing Scene 2 indicatesall scenes illuminated by E2( )l .

The figure illustrates why constancy is impossible in general. Whenviewing Image 1, the visual system cannot tell whether Scene 1 orScene 1̃ is actually present: achromatic points measured for a testpatch embedded in these two scenes must be the same, even thoughthe scene illuminants are as different as they are for Scenes 1 and 2.Recall from the data analysis above that this result (no change ofachromatic point across a change of illuminant) indicates the absenceof constancy.

The figure also illustrates why constancy can be shown across somescene pairs. Scenes 1 and 2 produce distinguishable retinal images, sothere is no a priori reason that the measured achromatic points fortest patches embedded in these two scenes need bear any relation toeach other. In particular, there is no constraint that prevents thechange in achromatic points across the two scenes from tracking thecorresponding illuminant change. Indeed, one interpretation of thegood constancy shown by the data reported above is that the visualsystem infers approximately the correct illuminants for Scenes 1 and2. A mystery would only occur if it could also infer the correctilluminants for Scenes 1̃ and 2̃.


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E1( )l

E2( )l

Scene 1̃

Scene 1

Scene 2̃

Scene 2

Image 2

Image 1

Figure 5. Schematic illustration of the ambiguity inherentin color constancy. The figure shows, schematically, the setof all scenes. Each point in the schematic represents a possiblescene. Scenes 1 and 2 from the experiment described aboveare indicated by closed circles. Each hatched ellipse encloses asubset of scenes that all produce the same image. The scenesrepresented by open circles, Scenes 1̃ and 2̃, produce the sameimages as Scenes 1 and 2 respectively. The open ellipses eachenclose a subset of scenes that share the same illuminant.

Figure 6 replots the results from achromatic measurements madefor Scene 1 together with the results for a new Scene, ˜̃1. Theilluminant in Scene ˜̃1 is the same as that in Scene 2, but the objects inthe scene have been changed to make the image reflected to the eyefor Scene ˜̃1 highly similar to that reflected for Scene 1; Scene ˜̃1 is anexperimental approximation to the idealized Scene 1̃ described above.It would be surprising indeed if constancy were good when assessedbetween Scenes 1 and ˜̃1, and it is not. The achromatic points


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measured for Scenes 1 and ˜̃1 are very similar, with the constancyindex between them being 0.11.

0.2

0.3

0.4

0.5

0.2 0.3 0.4 0.5

IlluminantAchromaticConstancy

CIE

y c

hrom

atic

ity

CIE x chromaticity

Scene 1 Scene 1~~

Figure 6. Achromatic data when both illuminant andscene surfaces are varied. The images at the top of the

figure show the observer’s view of two scenes, labeled 1 and ˜̃1.The relation between these scenes is described in the text. Thetest patch is visible in each image. The projection colorimeterwas turned off at the time the images were acquired, so theimages do not show the results of observers’ achromaticadjustments. The chromaticity diagram shows the data fromachromatic adjustments of the test patch made in the context ofthe two scenes. Same format as Figure 4. The equivalentsurface reflectance ˜( )S l computed from the data obtained inScene 1 is shown in Figure 4.


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Discussion

Constancy depends on image ensemble studied

The analysis and data presented above show that the degree ofhuman color constancy depends on the choice of scenes across whichit is assessed. Thus it is not useful to summarize human performancethrough blanket statements about the degree of constancy obtained.Rather, questions about constancy must be framed in conjunction witha specification of scene ensemble. Natural questions are a) whatensembles of scenes support good constancy? and b) how doesconstancy vary within some ensemble of scenes which is intrinsically ofinterest? An example of the latter would be scenes that occur innatural viewing.

The choice of experimental scenes is a crucial aspect of the designof any constancy experiment. Without some a priori restriction, thenumber of possible scenes is astronomical, and systematic explorationof the effect of all possible stimulus variables is not feasible. Inchoosing an ensemble of scenes for study, different experimentershave been guided by different intuitions. Indeed, it is this choice thatmost differentiates various studies. A common rationale, however, isto test specific hypotheses about how constancy might operate. Thegoal is to develop principles that allow generalization beyond thescenes studied experimentally.

Two broad approaches have been pursued. The mechanisticapproach is based on the hope that constancy is mediated by simplevisual mechanisms, and that these mechanisms can be studiedthrough experiments with simple stimuli (e.g. uniform test patchespresented on uniform background fields.) The computational approachis to develop image processing algorithms that can achieve colorconstancy and to use insight gained from the algorithms to buildmodels of human performance. This approach is often characterizedby the use of stimuli closer to those encountered in natural viewing, asthe algorithms are generally designed to take advantage of thestatistical structure of natural images. The difference between themechanistic and computational approaches is not always clear cut – amechanistic theory that explains human constancy can always berecast as a computational algorithm, while the action of a givenalgorithm can probably be approximated by the action of a series ofplausible neural mechanisms (see e.g. Marr, 1982, Chapter 1).Examples of both approaches are outlined below.


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The mechanistic approach

Constancy is essentially a relative phenomenon – it can only beassessed by measuring appearance across two (or more) scenes. Wecannot say from the data above that constancy is good for Scenes 1and 2 but bad for Scene ˜̃1. Rather, constancy is good across thechange from Scene 1 to Scene 2 but bad across the change fromScene 1 to Scene ˜̃1. Presumably it is possible to construct some otherScene 3 such that good constancy is revealed across Scenes ˜̃1 and 3.

What is it about the relation between Scenes 1 and 2 that supportsthe good constancy observed? A critical feature is that all that differsbetween them is the spectrum of the ambient illuminant. This designis common to most studies of constancy -- stability of appearance isassessed under conditions where the surfaces comprising the sceneare held fixed while the illuminant is varied (e.g. Helson & Michels,1940; Burnham, Evans, & Newhall, 1957; McCann, McKee, & Taylor,1976; Arend & Reeves, 1986; Breneman, 1987; Brainard & Wandell,1992). It is probably the ubiquity of this surfaces-held-fixed designthat leads to the oft-quoted generalization that human vision isapproximately color constant (e.g. Boring, 1942).

When the surfaces in the image are held constant, it is easy topostulate mechanisms that could, qualitatively at least, support thehigh levels of observed constancy.

The initial encoding of the color signal by the visual system is theabsorption of light quanta by photopigment in three classes of conephotoreceptors, the L-, M-, and S-cones (for a fuller treatment see e.g.Rodieck, 1998; Kaiser and Boynton, 1996; Brainard, 1995). The threeclasses are distinguished by how their photopigments absorb light as afunction of wavelength. The fact that color vision is based onabsorptions in three classes of cones is the biological substrate fortrichromacy.

An alternative to using tristimulus or chromaticity coordinates torepresent spectral properties of the light reaching the eye is to usecone excitation coordinates. These are proportional to the quantalabsorption rates for the three classes of cones elicited by the light.The cone excitation coordinates for a light, r, may be specified byusing a three-dimensional column vector

r =È

Î

ÍÍ

˘

˚

˙˙

r

r

r

L

M

S

. (4)


18

It is well-accepted that the signals initiated by quantal absorptionare regulated by adaptation. A first-order model of adaptationpostulates that a) the adapted signals are determined from quantalabsorption rates through multiplicative gain control, b) at each retinallocation the gains are set independently within each cone class, so that(e.g.) signals from M and S cone do not influence the gain of L cones,and c) for each cone class the gains are set in inverse proportion to aspatial average of the quantal absorption rates seen by cones of thesame class. This model is generally attributed to von Kries (1905).The three postulates together are sometimes referred to as von Kriesadaptation. Von Kries recognized that models where some of thepostulates hold and others do not could also be considered.

The first postulate of von Kries adaptation asserts that for eachcone class, there is an adapted cone signal ( aLfor the L-cones, aM forthe M-cones, and aS for the S-cones) that is obtained from thecorresponding cone excitation coordinate through multiplication by again (e.g. a g rL L L= ). The may be expressed using the vector notationintroduced in Eqn. 4. Let the vector a represent the magnitude of theadapted cone signals. Then

a D r=È

Î

ÍÍ

˘

˚

˙˙

=È

Î

ÍÍ

˘

˚

˙˙

È

Î

ÍÍ

˘

˚

˙˙

=a

a

a

g

g

g

r

r

r

L

M

S

L

M

S

L

M

S

0 0

0 0

0 0

. (5)

Because the adapted cone signals a are obtained from the coneexcitation coordinates r through multiplication by the diagonal matrixD, this postulate is called the diagonal model for adaptation. It shouldbe emphasized that for the diagonal model to have predictive power,all of the effect of context on color processing should be captured byEqn. 5. Under this model two test patches that have the sameadapted cone signals should have the same appearance.

In general, it is conceptually useful to separate two components ofa model of adaptation (Krantz, 1968; Brainard & Wandell, 1992). Thefirst component specifies what parameters of a visual processingmodel are allowed to vary with adaptation. The diagonal modelprovides this component of the full von Kries model. Under thediagonal model, the only parameters that can vary are the three gains.

The second component of a full model specifies how the processingparameters are determined by the image. The diagonal model is silentabout this, but the issue is addressed by the second two assumptionsof the full von Kries model. Only cone excitation coordinates within acone class influence the gain for that cone class, and the specific form


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of the influence is that the gain at a location is set inverselyproportional to the mean excitation in a neighborhood of the location.

If the visual system implements von Kries adaptation, the adaptedcone signals coding the light reflected from a surface are considerablystabilized across illuminant variation, provided that the other surfacesin the scene also remain fixed (Brainard & Wandell, 1986; Lennie &D’Zmura, 1988; Foster & Nascimento, 1994; see also Finlayson, Drew,& Funt, 1994). Indeed, von Kries adaptation is the active ingredient inlater versions of Land’s popular ‘retinex’ account for successive colorconstancy. In the descriptions of the retinex algorithm, the adaptedcone signals are called lightness designators, and the lightnessdesignators are derived from cone excitations through elaboratecalculation. None-the-less, for successive constancy the calculationreduces to a close approximation to classic von Kries adaptation (Land,1986; see Brainard & Wandell, 1986; for early descriptions of Land’swork see Land, 1959a; 1959b; Land & McCann, 1971).

Qualitatively, then, von Kries adaptation can explain the goodconstancy shown in experiments where the illuminant is changed andthe rest of the surfaces are held fixed. Such adaptation also providesa qualitative account for the poor constancy shown by the data inFigure 6, where both the illuminant and surfaces in the scene werechanged so as to hold the image approximately constant. On the basisof the data presented so far, one might sensibly entertain the notionthat human color constancy is a consequence of early adaptive gaincontrol.

Each of the postulates of von Kries adaptation have been subjectedto sharply focused empirical test, and it is clear that each fails whenexamined closely. With respect to the diagonal model, a number ofexperimental results suggest that there must be additional adaptationthat is not described by Eqn 5. These effects include gain control atneural sites located after signals from different cone classes combineand signal regulation characterized by a subtractive process ratherthan multiplicative gain control (e.g. Hurvich & Jameson, 1958;Jameson & Hurvich, 1964; Walraven, 1976; Shevell, 1978; Poirson &Wandell, 1993; see Eskew, McLellan & Giulianini 1999; Webster,1996).

The diagonal model fails when probed with stimuli carefully craftedto test its assumptions. Does it also fail for natural scenes? Theilluminant spectral power distributions E( )l and surface spectralreflectance functions S( )l found in natural scenes are not arbitrary.Rather, these functions tend to vary smoothly as a function ofwavelength. This constraint restricts the range of color signals likely


20

to occur in natural scenes. Conditions that elicit performance incontradiction to the diagonal model may not occur for natural scenes.If so, the diagonal model would remain a good choice for studies ofhow adaptive parameters vary within this restricted domain.

The regularity of illuminant and surface spectral functions may becaptured with the use of small-dimensional linear models (e.g. Cohen,1964; Judd, MacAdam, & Wyszecki, 1964; Maloney, 1986;Jaaskelainen, Parkkinen, & Toyooka, 1990). The idea of a linear modelis simple. The model is defined by N basis functions. These are fixedfunctions of wavelength, E EN1( ), , ( )l lL . Any spectrum E( )l isapproximated within the linear model by a weighted sum of the basisfunctions,

˜ ( ) ( ) ( )E w E w EN Nl l l= + +1 1 L (6)

where the weights are chosen to minimize the approximation error.Figure 7 plots the basis functions of a three-dimensional linear modelfor natural daylights, and shows the linear model approximation to ameasured daylight. The same approach can be used to expressconstraints on surface reflectance functions. The use of linear modelshas been central to computational work on color constancy (seeMaloney, 1999; Brainard & Freeman, 1997; Brainard, Kraft, &Longère, in press).

Mondrian World scenes where the illuminants and surface spectraare restricted to be typical of naturally occurring spectra can bereferred to as Restricted Mondrian World scenes. When theexperimental scenes are from the Restricted Mondrian World, thediagonal model seems to provide a good description of performance.


21

400 450 500 550 600 650 7000

0.5

1

Pow

er (

arbi

trar

y un

its)

Wavelength (nm)

400 450 500 550 600 650 700-1

0

1

Pow

er (

arbi

trar

y un

its)

Wavelength (nm)

400 450 500 550 600 650 700-1

0

1

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er (

arbi

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its)

Wavelength (nm)

400 450 500 550 600 650 7000

0.5

1

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Wavelength (nm)

Figure 7. Linear model for natural daylights. Top) Threebasis functions for the CIE linear model for daylights (CIE,1986). Bottom) Linear model approximation to a measureddaylight. Solid line, measurement. Dotted line, reconstruction.The measured daylight was obtained from a database ofmeasurements made available by J. Parkkinen and P. Silfsten onthe world-wide web at URLhttp://cs.joensuu.fi/~spectral/databases/download/daylight.htm

Brainard & Wandell (1992) tested the diagonal model usingasymmetric matching. In asymmetric matching, the observer adjustsa matching patch, embedded in one scene, so that its appearancematches that of a test patch presented in another. In the context ofthe diagonal model, the match is taken to indicate two lights that elicit


22

identical adapted cone signals. Let a t and am represent the adaptedcone signals for the test and matching patches. Eqn 5 then yields

a D r a D r r D D r D rt t t m m m m m t t t m t= = = fi = [ ] [ ] =- Æ1

, (7)

where the diagonal matrix

Dt m

g

g

g

g

g

g

L

t

L

m

M

t

M

m

S

t

S

m

Æ =

È

Î

ÍÍÍÍÍ

˘

˚

˙˙˙˙˙

0 0

0 0

0 0

. (8)

If the diagonal model is correct, the cone excitation coordinates of thetest and match patches are related by a diagonal matrix whose entriesare the ratios of cone gains. A single asymmetric match determinesthe entries of the matrix Dt mÆ . Since Eqn. 7 must hold with the samematrix Dt mÆ for any choice of test patch cone coordinates, repeatingthe experiment with different test patches allows evaluation of thediagonal model.

In a successive matching experiment that employed simplesynthetic scenes, Brainard & Wandell (1992) found that a large set ofasymmetric matching data was in good agreement with the diagonalmodel. Similar results were obtained by Bauml (1995, also forsuccessive matching) and by Brainard, Brunt, & Speigle (1997, forsimultaneous matching). To develop a theory of color constancy thatapplies to natural viewing, violations of the diagonal model may besmall enough to neglect.

What about the other postulates of von Kries adaptation, whichconcern how the gains are set by image context? The notion that thegains are set as a function of a spatial average of the image has beentested in a number of ways. One approach is to examine theequivalent background hypothesis. In its general form, this hypothesisasserts that the effect of any image on the color appearance of a testlight is the same as that of some uniform background. If the gains areset by a spatial average of the image, then the equivalent backgroundhypothesis must also hold.

Precise tests of the equivalent background, where spatial variationof color is introduced into a uniform background, indicate that it fails(Brown & MacLeod, 1997; Shevell & Wei, 1998). The general logic isfirst to find a uniform field and a spatially variegated field that haveidentical effect on the appearance of a single test light, and then showthat for some other test light these two contexts have different effects


23

(see Stiles & Crawford, 1932). As with the sharpest tests of thediagonal model, however, these studies did not employ stimuli fromthe Restricted Mondrian World.

Kraft & Brainard (1999) examined whether the spatial mean of animage controls the state of adaptation by constructing two illuminatedscenes that had the same spatial mean. To equate the mean, both theilluminant and the scene surfaces were varied between the scenes.Kraft & Brainard (1999) then measured the achromatic loci in the twoscenes. If the spatial mean were the only factor controllingadaptation, then the achromatic point in the two scenes should havebeen the same. A more general constancy mechanism, however,might be able to detect the change in the illuminant based on someother aspect of the images. The achromatic points were distinct, witha mean constancy index of 0.39. Even for nearly natural scenescontrol of the gains is not a simple function of the spatial mean of theimage: the visual system has access to additional cues to theilluminant. Kraft & Brainard (1999) examined other simple hypothesesabout control of adaptation in nearly natural scenes and found thatnone accounted for the data.

A key feature in the design of Kraft & Brainard’s (1999; see alsoKraft, Maloney, & Brainard, 2002; McCann, 1994; Gilchrist & Jacobson,1984) is that both the illuminant and the surfaces in the scene werevaried. When only the illuminant is varied, the data are roughlyconsistent with adaptation to the spatial mean of the image. Suchdata do not provide a sharp test, however, since essentially allplausible hypotheses predict good constancy when the surfaces in theimage are held fixed across an illuminant change. Only by varying thesurfaces and illuminants to differentiate predictions can strong tests bemade. This point also applies to studies of the neural locus ofconstancy.

The current state of affairs for the mechanistic approach may besummarized roughly as follows. A gain control model provides areasonable approximation to performance measured in scenesconsisting of illuminated surfaces, but lacking is a theory that links thegains to the image. The agenda is to understand what image factorscontrol the state of adaptation. In the lightness literature, this issometimes referred to as the ‘anchoring problem’ (e.g. Gilchrist et al.,1999). Within the mechanistic approach, one recent theme is to studythe influence of image contrast (Krauskopf, Williams, & Healey, 1982;Webster & Mollon, 1991; Singer & D’Zmura, 1995; Brown & MacLeod,1997; Shevell & Wei, 1998; Golz & MacLeod, 2002) and spatialfrequency content (Poirson & Wandell, 1993; Bauml & Wandell, 1996).Another approach (Brainard & Wandell, 1992; Bauml, 1995;


24

Chichilnisky & Wandell, 1995) is to study rules of combination (e.g.linearity) that allow prediction of parameter values for many imageson the basis of measurements made for just a few.

The computational approach

The mechanistic approach is motivated by consideration of thephysiology and anatomy of the visual pathways. The computationalapproach begins with consideration about how one could, in principle,process the retinal image to produce a stable representation of surfacecolor. The computational approach focuses on the informationcontained in the image, rather than on the specific operation ofmechanisms that extract the information.

Computational algorithms often operate in two distinct steps(Maloney, 1999). The first step estimates the illuminant at eachimage location, while the second step uses the estimate to transformthe cone coordinates at each location to an illuminant-invariantrepresentation. Given linear model constraints on natural surfacereflectance functions, the second step is quite straightforward(Buchsbaum, 1980) and is well-approximated by the diagonal gaincontrol (Brainard & Wandell, 1986; Foster & Nascimento, 1994). Thedeep issue is what aspects of the image carry useful information aboutthe illuminant. This issue is completely analogous to the central issuewithin the mechanistic approach, namely what aspects of the imagecontrol adaptation. Indeed, the idea linking the computationalalgorithms to measured human performance is that measuredadaptation might be governed by the same image statistics thatprovide information about the illuminant (Maloney, 1999; Brainard,Kraft, and Longère, in press).

Many algorithms have been proposed for estimating the illuminantfrom the image (Buchsbaum, 1980; Maloney & Wandell, 1986; Lee,1986; Funt & Drew, 1988; Funt, Drew & Ho, 1988; Forsyth, 1990;D’Zmura & Iverson, 1993; D'Zmura, Iverson, & Singer, 1995; Brainard& Freeman, 1997; Finlayson, Hubel, & Hordley, 1997). A detailedreview of the individual algorithms is beyond the scope of this chapter,but excellent reviews are available (Hurlbert, 1998; Maloney, 1999).Common across algorithms is the general approach of specifyingassumptions that restrict the class of scenes and then showing how itis possible to estimate the illuminant within the restricted class. Withreference to Figure 5 above, each algorithm is based on a rule forchoosing one particular scene from within each hatched ellipse.

In practice, different proposed algorithms depend on differentimage statistics. For example, in Buchsbaum’s (1980) classic


25

algorithm the illuminant estimate was based on the spatial mean ofthe cone quantal absorption rates. As a model for humanperformance, this algorithm may be tested by asking whetheradaptation is governed only by the spatial mean. As described above,experiments show that this is not the case. The detailed logicconnecting this algorithm to human performance is described in arecent review (Brainard, Kraft, and Longère, in press).

Other computational algorithms depend on different aspects of theimage. For example, Lee (1986; also D’Zmura and Lennie, 1986)showed that specular highlights in an image carry information aboutthe illuminant. This has led to tests of whether human vision takesadvantage of the information contained in specular highlights(Hurlbert, Lee, & Bulthoff, 1989; Yang & Maloney, 2001).

In Yang & Maloney’s work (2001), the stimuli consisted of realisticgraphics renderings of synthetic scenes. That is, the locations,spectral properties, and geometric properties of the scene illuminantsand surfaces were specified in software, and a physics-based renderingalgorithm was used to generate the stimuli. In real scenes, theinformation provided by separate cues tends to covary, which makes itdifficult to separate their effects. By using synthetic imagery, Yang &Maloney teased apart effects of independent cues. They were ableshow that specular highlights can influence human judgments ofsurface color appearance and to begin to delineate the circumstancesunder which this happens. Delahunt (2001) employed similartechniques to study the role of prior information about naturaldaylights in successive color constancy. (For computational analysis ofthe use of such information, see D'Zmura, Iverson, & Singer, 1995;Brainard & Freeman, 1997.) The methodology promises to allowsystematic study of a variety of hypotheses extracted from thecomputational literature.

Generalizing to simultaneous constancy

This chapter has focused on successive color constancy, and inparticular on the case where the illuminant is approximately uniformacross the scene. As illustrated by Figure 2, this idealized situationdoes not hold for natural scenes.

When an image arises from a scene with multiple illuminants, onecan still consider the problem of successive color constancy. That is,one can ask what happens to the color appearance of an object in thescene when the spectral properties of the illuminant are changedwithout a change in scene geometry. Little, if any, experimental efforthas been devoted to this question.


26

The presence of multiple illuminants within a single scene alsoraises the question of simultaneous constancy – how similar does thesame object appear when located at different places within the scene?

One thread of the literature has emphasized the role of scenegeometry (Hochberg & Beck, 1954; Epstein, 1961; Flock & Freedberg,1970; Gilchrist, 1977; 1980; Knill & Kersten, 1991; Pessoa, Mingolla,& Arend, 1996; Bloj, Kersten, & Hurlbert, 1999; Bloj & Hurlbert,2002). Under some conditions, the perceived orientation of a surfacein a scene can influence its apparent lightness and color, in a mannerthat promotes constancy. The range of conditions under which thishappens, however, is not currently well-understood.

An interesting aspect of simultaneous constancy is that observerperformance can depend heavily on experimental instructions. In astudy of simultaneous color constancy, Arend & Reeves (1986) hadobservers adjust the color of one region of a stimulus display until itappeared the same as another. They found that observers’ matchesvaried with whether they were asked to judge the color of the reflectedlight or the color of the underlying surface. More constancy wasshown when observers were asked to judge the surface (see alsoBauml, 1999; Bloj & Hurlbert, 2002). In a study of successiveconstancy, on the other hand, Delahunt (2001) found only a smallinstructional effect. It is not yet clear what conditions support theinstructional dichotomy, nor whether the dichotomy indicates dualperceptual representations or observers’ ability to reason fromappearance to identity.

Recent theories of lightness perception have emphasizedsimultaneous constancy (Gilchrist et al., 1999; Adelson, 1999). At thecore of these theories is that idea that perception of lightness (andpresumably color) proceeds in two basic stages. First, the visualsystem segments the scene into separate regions. Second, imagedata within regions is used to set the state of adaptation for thatregion. (At a more detailed level, the theories also allow for someinteraction between the states of adaptation in different regions.) Thetwo-stage conception provides one way that results for successiveconstancy might generalize to handle simultaneous constancy: modelsthat explain successive constancy for uniformly illuminated scenesmight also describe the processes that set the state of adaptationwithin separately segmented regions within a single image (seeAdelson, 1999). To the extent that this hypothesis holds, it suggeststhat work on simultaneous constancy should focus on thesegmentation process. At the same time, it must be recognized thatthe segment-estimate hypothesis is not the only computationalalternative (see e.g. Land & McCann, 1971; Funt & Drew, 1988;


27

Adelson & Pentland, 1996; Zaidi, 1998) and that empirical tests of thegeneral idea should also be given high priority.

Acknowledgments

I thank P. Delahunt, B. Wandell, and J. Werner for discussion andfor comments on draft versions of this chapter. Supported by NEIgrand EY 10016.

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