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R. Tokunaga and A. D. Logvinenko Vol. 27, No. 12 /December 2010 /J. Opt. Soc. Am. A 2551

Hue manifold

Rumi Tokunaga1,* and Alexander D. Logvinenko2

1Research Institute of Electrical Communication, Tohoku University, Katahira Aoba, Sendai 980-8577, Japan2Department of Vision Sciences, Glasgow Caledonian University, City Campus, Cowcaddens Road,

Glasgow G4 0BA, UK*Corresponding author: rumi.tokunaga@riec.tohoku.ac.jp

Received March 29, 2010; revised August 25, 2010; accepted August 25, 2010;posted September 28, 2010 (Doc. ID 126191); published November 5, 2010

It is generally accepted that hues can be arranged so as to make a circle. The circular representation of hue hasbeen supported by multidimensional scaling, which allows for the representation of a set of colored papers asa configuration in a Euclidean space where the distances between the papers correspond to the perceptualdissimilarities between them. In particular, when papers of various hues are evenly illuminated, they are ar-

ranged in a one-dimensional circular configuration. However, under variegated illumination we show that thesame papers in fact make a two-dimensional configuration that resembles a torus. © 2010 Optical Society ofAmerica

OCIS codes: 330.0330, 330.1720.

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. INTRODUCTIONhere are two ways to alter the color of an object. One canhange either the object’s surface reflectance or the ob-ect’s illumination. For instance, one can make a whiteurface look yellowish either painting it with yellow paintr using an incandescent light source. Will it be the sameellow? Here, we report on experiments that suggest thathese are two different yellow hues. In other words, we ar-ue that the hue palette induced by changing the mate-ial properties of a surface differs from that induced byhanging the surface lighting. This challenges a majorogma of color science: all hues make a one-dimensionalanifold topologically equivalent to a circle.In fact, there is as much experimental evidence against

his dogma as in favor of it. On one hand, it has been es-ablished that human observers readily distinguish be-ween lighting- and material-induced object color alter-tions [1–3], although it is hardly surprising, because asnown from everyday life, we are unlikely to mistake col-red shadows for material stains. If against the odds thisappens, we call it a color illusion [4–6]. Such an abilityould be impossible if the color palettes induced by ma-

erial and light are identical. On the other hand, there isstrong belief that both of the hue palettes have the same

ircular structure. Since Newton’s time it has been takenor granted that the lights of all the possible hues form ahue circle.” Also, it has been known for centuries that pa-ers of various hues can also be arranged in a hue circlesee for a comprehensive historical review [7]). Further-

ore, when observers are asked to place a set of coloredapers on a table so that the distances between them areroportional to their subjective proximities, the typical re-ult is a hue circle.

Representing subjective proximities between objects asistances between points in a Euclidian space of minimalimensionality is the key idea of multidimensional scal-ng (MDS). Particularly, non-metric MDS is a technique

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hich takes a matrix of dissimilarities between a numberf objects and creates a configuration of points, such thatf the proximity between a pair of objects a and b is notess than that between c and d, then the distance betweenhe points representing a and b is not less than that rep-esenting c and d [8–10]. MDS analysis corroborates thentuitive conjecture concerning the circular arrangementf the hue palette of lights [9,11,12] and colored paperssee [13]).

It follows that when we present various colored papersit by daylight against a white background their hues wille arranged in a hue circle. Likewise, when we displayarious patches of colored lights on the same white back-round their hues will be also arranged in a hue circle.re these two (i.e., material and lighting) hue circles theame or different? What happens when both the papersnd lights are varied? How will the hues be arranged inhis case? We address these questions experimentally us-ng MDS. It should be mentioned that, to our knowledge,

DS has never been employed before to ascertain the di-ensionality of the hue palette for variegated scenes withultiple illuminants.As non-metric algorithms of MDS imply that the dis-

ances between colors are in the same order as the corre-ponding dissimilarities, an ideal method of collectingata would be quadruple comparisons—that is, compar-ng all pairs of pairs of stimulus colors. In other words, a

atrix of quadruple comparisons should, in principle, besed as the input to non-metric MDS algorithms. How-ver, as the method of quadruple comparison is ratherime-consuming, experimenters resort to more rapid tech-iques to quantify color dissimilarities, such as ratingingle pairs [9,11,14–19]. Rating is particularly efficientecause it reduces quadruple comparisons to pair com-arisons, thereby decreasing the number of evaluations.s the number of pairs to be assessed in our experimentas large, we also chose to use numerical rating to evalu-

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2552 J. Opt. Soc. Am. A/Vol. 27, No. 12 /December 2010 R. Tokunaga and A. D. Logvinenko

te dissimilarities. A preliminary experiment was carriedut to check whether quadruple comparisons and numeri-al rating both produce the same results.

. METHODS. Stimulihe experimental setup was analogous to that used pre-iously [19]. The experimental display consisted of sixectangular fields made of white paper with blackandom-dot design (Fig. 1). Two fields (1 and 2 in Fig. 1)ere used to present the stimulus papers—a sample from

he Munsell Book of Color [20]. The Munsell notation ofhe stimulus papers was 5R4/14, 5YR7/12, 5Y8/12, 5G6/0, 10BG5/8, 5PB5/12, and 10P5/12. The distance be-ween the display and the observers was 2 m. The angu-ar size of the whole display was 26° �29°, with eachaper subtending an angle of approximately 4°. The ex-erimental room was semidarkened. The vision was bin-cular.

In the preliminary experiment the entire experimentalisplay was evenly illuminated by a neutral light (CIE931 chromaticity coordinates x=0.397, y=0.411, 110 lx).n the first part of the preliminary experiment all theelds except for fields 1 and 2 were left empty. In the sec-nd part of the preliminary experiment and in the mainxperiment fields 3 and 4 were used to present the stan-ard pair of the stimulus papers (see below), with fields 5nd 6 being empty.In the main experiment, fields 1 and 2 were illumi-

ated independently by one of five lights (yellow, blue,ed, green, and purple), giving a total of 15 possible illu-ination conditions (red-red, red-yellow, etc.). A digital

rojector (NEC MT1050) was used to control the illumi-ation. The lights were approximately equiluminant with

ig. 1. (Color online) Experimental setup and stimulus display.digital projector (DP) provided the independent illumination of

he six fields of the stimulus display (SD), these being covered byhite paper with a random-dot design. Seven Munsell papersere presented in the fields 1 and 2. Fields 3 and 4 were used toisplay the pair of Munsell papers that represented standard dis-imilarity. Fields 5 and 6 were illuminated so as to counter-alance the overall illumination. A computer (PC) randomlyurned on a pair of light-emitting diodes (one in field 1, and onen field 2), which indicated the pair of Munsell papers to bevaluated. An observer entered the dissimilarity estimate byressing a button on the response box (RB).

IE 1976 u�v�-coordinates (0.382, 0.488), (0.199, 0.530),0.127, 0.532), (0.183, 0.210), and (0.259, 0.365). The illu-inance fell in the range 59–66 lx. Figure 2(a) shows theIE 1976 u�v�-coordinates of the 35 lights reflected fromll seven of the Munsell papers under each of the five il-uminations. Fields 5 and 6 empty of Munsell papers wereet up to balance the overall illumination in the experi-ental display. Their illumination was chosen (from theve lights) so as to reduce chromatic adaptation.The fields in the experimental display were clearly

eparated from each other with black thin dividers. Theyere illuminated so that there were no illumination bor-ers between spotlights. When the lights were different,ll the observers had clear understanding that the fieldsre differently illuminated. Specifically, they saw real col-red papers lit by real lights of different colors.

. Procedures of the Preliminary Experimentn the first part of the preliminary experiment quadrupleomparisons were used as the judgment method. The ex-erimenter pointed out at random two different pairs ofifferent papers (one in field 1 and the other in field 2),sking in which pair the papers were more dissimilar inolor. Each pair of pairs (210 in total) was evaluated tenimes by each observer. The number of times the pair wasreferred to all other pairs as most dissimilar was takens the measure of dissimilarity between papers (for moreetails, see [21]).Then, in the second part of the preliminary experiment,

he method of rating was used for judgment. Two differentapers in different fields were singled out randomly byhe experimenter. A pair of yellow (5Y8/12) and blue10PB5/12) papers was set up as a standard of dissimilar-ty. It was permanently present in the observer’s viewfields 3 and 4 in Fig. 1). Observers were asked to evalu-te the dissimilarity between the papers as compared tohe standard pair using a number, taking the standard as00. Each pair was evaluated five times by each observer.ive normal trichromatic observers took part in the pre-

iminary experiment.

. Procedures of the Main Experimentn the main experiment, the method of rating was used asudgment. The experiment was divided into sessions. Inach session a fixed pair of chromatic lights was used. Aession consisted of 49 trials in which all the 7�7 pairs ofapers were evaluated. In each trial two papers in differ-nt fields were indicated randomly by light-emitting di-des next to each paper. Observers were first askedhether they looked the same or different. All the observ-rs found the papers to appear different. (Admittedly, theame papers lit by the same light looked alike.) Then,hey were asked to evaluate the dissimilarity between theapers as compared to the standard, taking as the stan-ard the yellow paper (5Y8/12) under the yellow illumina-ion and the blue paper (10PB5/12) under the blue illumi-ation (fields 3 and 4 in Fig. 1). Each session wasepeated six times for each observer.

Out of the five observers who took part in the prelimi-ary experiment, two went on to take part in the main ex-eriment. They completed 15 sessions (i.e., for all possibleombinations of five illuminations). We also used a control

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R. Tokunaga and A. D. Logvinenko Vol. 27, No. 12 /December 2010 /J. Opt. Soc. Am. A 2553

roup of three more observers, who conducted only tenessions (all the combinations of the yellow, blue, red, andreen illuminations). Results for all the observers wereimilar. Only the data obtained for the two observers whoarried out all 15 sessions are presented in this report.

. RESULTShe dissimilarities obtained in both parts of the prelimi-ary experiment were averaged separately across ses-ions and the five observers who took part in the prelimi-ary experiments. Then the averaged dissimilarityatrices were analyzed by using a non-metric MDS algo-

ithm. In order to establish how the output configurationsbtained by different methods related to each other, weonducted a Procrustes analysis [10], which performedhe permissible transformations (dilating, translating, re-ecting, and rotating) on the rating-based configurationo bring it as close as possible to the quadruple-omparison-based configuration. The transformed con-gurations, along with the output configurations pro-uced by quadruple comparisons, are shown in Fig. 3. Therocrustes statistic [[10], p. 127] has been evaluated as.026, which was small enough to justify the use of theating method in the main experiment (the smaller thetatistic value, the better match). While it is not quite cir-ular, the circular order of the colors therein is the sames in the Munsell book. A similar somewhat distortedhue circle” emerged in many early MDS studies of Mun-ell papers (e.g., [13,22,23]). The output configuration inig. 3 will be referred to as the material-hue contour.The dissimilarities collected in the main experiment

ere averaged over six repeats (i.e., sessions) and two ob-ervers. A resultant (average) matrix �35�35� of dissimi-arities was entered into the same MDS algorithm. Figure

shows a three-dimensional (3D) non-metric MDS solu-ion. Stress (i.e., an index showing the relative proportionf mismatches between the distances and the dissimilari-ies) was evaluated for two-dimensional (2D) and 3D out-

ig. 2. (Color online) Chromaticity of the lights reflected from976 uniform chromaticity diagram. The shape and face color oup)—5YR7/12, triangle (pointing right)—5Y8/12, square—5iamond—10P5/12.] The edge color of the markers specifies the coonnected with a line whose color denotes the color of the illuminifferent illuminants are connected by a line whose color resemb

ut space as 0.12 and 0.09, respectively. The smalleralue for the 3D space indicates that the 3D output con-guration better renders the dissimilarity structure. Atandard assumption underlying non-metric MDS of colortimuli is that observers’ dissimilarity judgments areonotonically related to the subjective distance between

he stimuli [24]. Numerous previous MDS studies onunsell papers homogeneously illuminated [13] corrobo-

ate this assumption. Hence, it is safe to believe thatig. 4 represents (up to some unknown order-preserving

ransformation) the configuration the stimulus papersake in the subjective color space.In Fig. 4 the points representing the same paper illu-inated by different lights [connected by a line of the

ame color as in Fig. 2(b)] make a closed contour which islled with a color similar to the color of the paper underaylight. These contours will be referred to as lighting-ue contours since they show how a paper’s position in the

nsell papers under various illuminations presented in the CIEarkers specify the color of the paper. [Circle—5R4/14, triangle

, pentagram—10BG5/8, triangle (pointing down)—5PB5/12,he illumination. (a) The Munsell papers lit by the same light are(b) The points representing the same Munsell paper under fivecolor of the paper under daylight.

ig. 3. (Color online) MDS output configurations (material-hueontours) obtained in the preliminary experiment. Theuadruple-comparison-based configuration is represented by theointed-up colored symbols. The transformed rating-based con-guration is marked by crosses. Notation of colored symbols ishe same as in Fig. 2.

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issimilarity space alters when the paper’s illuminationaries. The lighting-hue contours are arranged along alosed horizontal contour represented by a gray filling inig. 4. This contour is made by the centroids of all the

ighting-hue contours (hexagrams). Note that its form islose to that in Fig. 3. Because of this, and also becausehe shift along the horizontal contour is induced by varia-

ig. 5. Dissimilarities versus chromaticity differences. In eachapers illuminated separately by one of the five lights: yellow (Yhromaticity difference between the reflected lights evaluated inrespectively, chromaticity differences) of pairs were normalized b5Y8/12, 5PB5/12) being used as the standard (see Fig. 1), generatbserver individually.

ig. 4. (Color online) (a), (b) Two views of the output configura-ion produced by the non-metric MDS algorithm obtained in theain experiment. Each point represents a Munsell paper illumi-

ated by a particular light. Notation is the same as in Fig. 2.

ions between papers, we believe that the horizontal con-our represents the material-hue contour.

As the configurations in Figs. 2(b) and 4 are rather dif-erent, the dissimilarities between papers are unlikely toe determined by the chromaticity differences betweenhe reflected lights (i.e., by the distances between the cor-esponding points in Fig. 2). Indeed, Fig. 5 shows that theissimilarity judgements (especially for different pairs ofights) are poorly correlated with the corresponding dis-ances in the CIE 1976 uniform chromaticity diagram.or example, the blue paper (5PB5/12) under the purple

ight and the red paper (5R4/14) under the blue light haveearly equal chromaticity coordinates (Fig. 2). Yet, theyere perceived as a great deal dissimilar. More specifi-

ally, the dissimilarity between these papers was as muchs 70% of the maximal dissimilarity (which was observedetween the red paper under the red light and the blueaper under the blue light). Therefore, in a multiple illu-inant scene the color of an object cannot be reduced to

he color of the light reflected from its surface.It is important to point out the transversality of the

ighting-hue and material-hue contours that can belearly seen in Fig. 4(b) where a different view of theame configuration as in Fig. 4(a) is presented. Theighting-hue contours in Fig. 4(b) apparently protrude up-ard and downward from the material-hue contour plane.herefore, whereas in Fig. 2 the “paper” chromaticity con-

ours [Fig. 2(a)] and the “light” chromaticity contoursFig. 2(b)] are located in the same plane, the material-huend lighting-hue contours in Fig. 4 are transversal toach other.

. DISCUSSIONhe present results are in accord with the previous stud-

es where the existence of the material and lighting di-ensions has been shown for object colors [18,19,25,26].

the vertical axis is the dissimilarity between a pair of Munsell(B), red (R), green (G), and purple (P); the horizontal axis is theof the CIE 1976 uniform chromaticity diagram. Dissimilaritiesissimilarity (respectively, the chromaticity difference) of the pairer the lighting conditions in question (i.e., Y-B, P-R, etc.) for each

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R. Tokunaga and A. D. Logvinenko Vol. 27, No. 12 /December 2010 /J. Opt. Soc. Am. A 2555

articularly, Logvinenko and Maloney [18] showed thatchromatic Munsell papers illuminated by neutral lightsf different intensities produced a 2D MDS output con-guration. This configuration comprised regular arc-like

ayers parallel to each other. The papers illuminated byhe same light located in the same layer. Different layersorresponded to different illuminations. It was concludedhat the dissimilarity along the layer reflected the percep-ual difference in lightness, whereas the dissimilaritycross the layers reflected the perceptual difference inurface-brightness. Lightness and surface-brightnessere understood as the material and lighting dimensionsf achromatic object colors [18].

It should be pointed out that observers in this experi-ent judged as most dissimilar two pairs: (i) the black pa-

er lit by the brightest light versus the white paper lit byhe dimmest light and (ii) the white paper lit by therightest light versus the black paper lit by the dimmestight. Interestingly, the papers in the first pair reflectedhe lights of practically equal luminance, whereas the lu-inance of the lights reflected by the papers in the second

air differed by a factor of 85. It follows that the dissimi-arity judgments in this experiments were not based onhe brightness of the reflected light. Thus, the surface-rightness is different from the reflected light brightness.A similar fan-like structure was discovered in the

ollow-up experiment with yellow, neutral, and blue Mun-ell papers [19]. It follows that the new dimension (i.e.,urface-brightness) that has emerged in these experi-ents is not something specific to the achromatic domain,

ut can be interpreted as the achromatic lighting dimen-ion common to the achromatic and chromatic object col-rs.

In a subsequent experiment we used yellow, neutral,nd blue lights to illuminate the same yellow, neutral,nd blue Munsell papers [25]. In fact it was a replicationf Logvinenko and Maloney’s study [18] except that thechromatic dimension was replaced with the yellow-blueimension. The seven papers were chosen so that the re-ected light chromaticities fell on the straight line (alonghe yellow-blue direction) in the CIE chromaticity dia-ram. Moreover, alternation over the chosen three illumi-ants did not violate the linearity of the paper chromatic-

ty loci. Specifically, the alternation of illuminationesulted in only a shift of the papers chromaticity alonghe same yellow-blue direction. The illuminants were ad-usted so that the reflected lights from (i) the yellow paperit by the blue light, (ii) the gray paper lit by the neutralight, and (iii) the blue paper lit by the yellow light wereractically metameric (that is, they have very close CIEhromaticity coordinates and luminance values).

The results were, in a sense, similar to those obtainedn the achromatic domain [18]. Specifically, the MDS out-ut configuration consisted of three slightly zigzaggingines shifted parallel each other in the direction transver-al to the lines. Equally illuminated papers lay on theame line. Therefore, it was concluded that each line rep-esented the material yellow-blue continuum. The trans-ersal direction was interpreted as representing the light-ng yellow-blue continuum. Notably, the dissimilarityetween the yellow paper lit by the blue light and the blueaper lit by the yellow light was found to be nearly maxi-

al despite the chromaticity difference between the lightseflected from these two papers being negligible. This in-icates that the lighting dimension revealed in this studyoes not correlate with the chromaticity of the reflectedight nor does it with the reflected light luminance [25].

It should be mentioned that only two hues—yellow andlue—were employed in the experiment [25]. Althoughhe existence of lighting hues had been proved, much re-ained to be investigated about the lighting-hue dimen-

ion. What actually varied along the lighting dimensioniscovered in this experiment was the lighting chroma forhe lighting yellow and blue hues. The next clear step waso vary independently the paper and light chromaticities.n [26] seven Munsell papers of different hues (same as inhe present study) and the yellow, neutral, and blue illu-inations were used. That experiment brought about

hree material-hue circles each similar to that in Fig. 3hifted approximately parallel to each other in the trans-ersal direction. However, as in the previous experiment25], this direction could also be interpreted as that ofighting chroma.

The rationale (and the novelty) of the present experi-ent was to use more illuminations of various chroma-

icities so as to allow a hypothetical lighting-hue dimen-ion, if any, to reveal itself. The fact that the newimension that emerged in the present experiment is bentnto a closed contour proves that it is most likely theighting-hue dimension. At any rate, it is neither theurface-brightness dimension nor the lighting chroma di-ension discovered in the previous experiments

18,19,25,26].Note that the lighting-hue contours were found to be

maller than the material-hue contours. This is in appar-nt contrast with Fig. 2 where the light chromaticity con-ours [Fig. 2(b)] are much larger than the paper chroma-icity contours [Fig. 2(a)]. More specifically, the averageength of the paper chromaticity contours in Fig. 2(a) haseen evaluated as 0.5, whereas the average length of theight chromaticity contours in Fig. 2(b) is 0.85. Therefore,n average, the paper chromaticity contours in Fig. 2(a)re shorter than light chromaticity contours in Fig. 2(b)y a factor of 1.71. On the other hand, in Fig. 4 the aver-ge length of the material-hue contours has been evalu-ted [27] as 71.7 versus the average length of the lighting-ue contours being 44.6. Therefore, on average, theaterial-hue contours in Fig. 4 are longer than the

ighting-hue contours by a factor of 1.61. This observationorroborates our the previous finding that the same dif-erence in the CIE coordinates produces a significantlyarger subjective difference in the material, as comparedo lighting, color dimensions [18,25,26]. This phenomenonas called the “lighting discounting” [25]. As we can see,

he lighting discounting has been observed in the presentxperiment as well. The ratio of the material versusighting-hue contour average lengths (i.e., 1.61) over theatio of the paper versus light chromaticity contour aver-ge lengths (i.e., 1 /1.71=0.58) is 2.75. It can be taken as auantitative index of the lighting discounting obtained inhe present study.

The existence of lighting color dimensions has rel-vance to the problem of simultaneous color constancy. In-eed, it follows from our results that contrary to the gen-

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2556 J. Opt. Soc. Am. A/Vol. 27, No. 12 /December 2010 R. Tokunaga and A. D. Logvinenko

ral belief that the color of an object remains constanthen its illumination varies [28–31], the object color doesary with illumination. Note, however, that in our experi-ent this light-induced color variation took place mainly

n the dimension transversal (i.e., along the lighting-hueontour) to the dimension along which the color alterationnduced by a material (reflectance) change occurs. Be-ause of such transversality, an asymmetric color matchs, generally, impossible—a fact pointed out before18,32,33]. Indeed, an asymmetric color match is under-ain by an implicit assumption that, by manipulating re-ectance, one can equate the color appearance of two ob-

ects under different illuminations. As one can see in Fig., this is impossible because given a lighting-hue shiftwithin the same lighting-hue contour), any change in re-ectance will only bring about an additional transversalhift (along the material-hue contour), thereby increasingissimilarity. It must be pointed out, however, that al-hough exact asymmetric color match is impossible, nu-erous experiments on color constancy testify that “ap-

roximate” across-illuminant matching can be readilychieved. This is reflected in such definitions of color con-tancy as “relative stability” (e.g., [34]) or “qualitativereservation” [35] of object color appearance despite illu-ination variation. However, the problem is what exactly

s meant by these. We argued elsewhere that there is ev-ry indication that it is the material color dimensionslong which the object color remains practically constanthen the illumination changes [36].At first glance, our results corroborate a belief, dating

ack to the 19th century, that one can see separately andimultaneously the color of an object and the color of thellumination [37,38]. In line with this, MacLeod recentlyuggested that one needed six numbers for a color stimu-us to be specified: the tristimulus values of the incidentight and the tri-stimulus values of the spectral reflec-ance [39]. Although it is true that the object color stimu-us can be specified with six numbers [40], this sextupletannot be separated into the two independent triplets ofhe tristimulus values (for lights and objects) because ofhe so-called metamer mismatching [40]. It has beenroved that both triplets constituting these six numbersre determined by a pair object/light, not object sepa-ately and light separately [40].

If the material-hue contours are taken to represent the

ponding chromaticity differences.

bject color, and the lighting-hue contours the illumina-ion color, then all the lighting-hue contours must be ofhe same form because they represent the colors of theame five lights. However, in reality they are rather dif-erent (Fig. 4). The difference between them cannot be re-uced to random fluctuations of observers’ responses. Fig-re 6 presents the pairs of identical papers lit by different

ights. The dissimilarity between the identical papers isnduced only by the illumination’s chromaticity differ-nce. Therefore, if lighting-hue were the hue of illumina-ion, then the dissimilarity between all the pairs of iden-ical papers under the same illumination condition (i.e.,ithin the same plate) should be approximately equal.owever, this is the case only for the yellow-purple illu-ination condition (the Y-P plate in Fig. 6). The Friedman

est (two-way non-parametric analysis of variance) per-ormed on a subset of data generated only by the pairs ofdentical papers showed a significant effect of paper ��6

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94.4, p�0.001�. Hence, lighting hue is not illuminationue. Both lighting and material hues are attributes of ob-

ect color.If lighting and material hues were separable dimen-

ions, and both the lighting and material hues could beepresented as a circle, then—ideally—the hue manifoldor object color could be a torus (i.e., the boundary surfacef a doughnut-like body). Although the configuration inig. 4 can be likened to a body with a doughnut-shapedurface, this “doughnut” is quite irregular. This testifiesgainst that lighting and material hues are separable.till, the set of all the hues makes a 2D manifold that cane considered as a bundle of the lighting-hue contours ors a bundle of the material-hue contours. However, thisanifold cannot be factored into two one-dimensional

ontours. Thus, the hue manifold for object color is essen-ially 2D and cannot be reduced to the Cartesian productf the two one-dimensional (lighting and material) hueontours.

It follows that any opaque object can have a variety ofues dependent on that object’s illumination. For ex-mple, there is the whole continuum of whites. A whiteurface under daylight renders only one shade of these.arying the illumination, one can produce white underellow illumination, white under blue illumination, andhe like: these constitute the lighting continuum ofhites. All these whites are readily seen as white. Like-

ig. 6. (Color online) Dissimilarities versus chromaticity differences for the pairs of identical papers. Colors of symbols and shapes arehe same as in Fig. 2. Figures show how the dissimilarities, produced only by the difference in illumination, correlate with the corre-

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R. Tokunaga and A. D. Logvinenko Vol. 27, No. 12 /December 2010 /J. Opt. Soc. Am. A 2557

ise, there is a continuum of blues. As pointed out above,e effortlessly discriminate a blue surface under yellow

ight from a yellow surface under blue light even thoughhey reflect lights of the same chromaticity [25]. This in-icates that lighting blue hue and material blue hue areualitatively different perceptual experiences (see moren this issue in [25]).

To summarize, when illumination is fixed, the set ofhromatic hues for object color is one-dimensional and cane geometrically represented as a slightly distorted circle.et under varying illumination, the object color huesake a somewhat distorted torus, being essentially 2D.

CKNOWLEDGMENThis work has been supported by Engineering and Physi-al Sciences Research Council (EPSRC), research grantP/C010353/1 (A. D. Logvinenko).

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