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c’isim Rrs. Vol. 14. pp. I 13 1140. Pergamon Press 1974. Printed m Great Bruin.
OPPONENT-PROCESS ADDITIVITY-I:
RED/GREEN EQUILIBRIA’
JAMESLARIMFX
Department of Psychology, Temple University, Philadelphia, Pa. 19122, U.S.A. DAVID H. KRANTZand CAROL M. CICERONE Department of Psychology, University of Michigan, 330 Packard Rd.. Ann Arbor. Mi. 48104. U.S.A.
(Received 7 January 1974)
Abstract-A red/green equilibrium light is one which appears neither reddish nor greenish (i.e. either uniquely yellow, uniquely blue, or achromatic). A subset of spectral and nonspectral red/green equilibria was determined for several luminance levels, in order to test whether the set of all such equilibria is closed under linear color-mixture operations. The spectral loci ofequilibrium yellow and blue showedeither no variation or visually insignificant varia- tion over a range of l-2 log,, unit. There were no trends that were repeatable across observers. We con- cluded that spectral red/green equilibria are closed under scalar multiplication; consequently they are in- variant hues relative to the Bezold-Briicke shift. The additive mixture of yellow and blue equilibrium wavelengths, in any luminance ratio, is also an equilibrium light. Small changes of the yellowish component of a mixture toward redness or greeness must be compensated by predictable changes of the bluish component of the mixture toward greenness or red- ness. We concluded that yellow and blue equilibria are complementary relative to an equilibrium white; that desaturation of a yellow or blue equilibrium light with such a white produces no Abney hue shift; and that the set of red/green equilibria is closed under general linear operations. One consequence is that the red/green chromatic-response function, measured by the Jameson-Hurvich technique of cancellation to equilibrium, is a linear function of the individual’s color-matching coor- dinates. A second consequence of linear closure of equilibria is a strong constraint on the class of combina- tion rules by which receptor outputs are recoded into the red/green opponent process.
INTRODUCTION
In color-matching the two physical manipulations of light are additive mixture (wavelength by wavelength summation of the two spectral energy-density func- tions) and scalar multiplication (insertion or removal of neutral density filters, i.e. multiplication of the spec- tral energy-density function by a constant). Lights will be denoted by a, b, c,... and the two physical manipu- lations will be denoted by @ and +. The additive mix- ture of a and h is denoted a @b; the multiplication of a by a scale factor r (> 0) is denoted t * a. The equiva- lence relation of metameric matching will be denoted by a - b. Grassmann’s laws (18534) include the invariance of metameric matching with respect to the operations Q and . More precisely: (1)ifa -. b, then ta 5 t+ b; (2) a -^ bifandonlyifa@c^ -^ b@c. A color theory must include not only the facts of metameric matching, but also those of color appear- ance. It is reasonable to ask whether there are equiva- lence relations different from -., based on color
’ This research was supported by NSF grant GB 8181 to the University of Michigan and by an NIH postdoctoral fel- lowship lo the senior author.
appearance, that also satisfy Grassmann’s invariance laws. Hering (1878) proposed that any hue can be de- scribed in terms of its redness or greenness and its yel- lowness or blueness. Moreover, red and green appear to be opposite poles of one aspect of hue, since one cannot experience both in a single color ; and the same holds for yellow and blue. These two bipolar aspects are independent: red can be experienced simul- taneously with either yellow or blue; and similarly for green. The quantitative investigation of Hering’s opponent-process theory began with an experiment by Jameson and Hurvich (1955). They measured the amount (in terms of intensity) of a standard light that had to be added to a spectral light to just cancel out the spectral light’s redness greenness, yellowness, or blueness. For example, to cancel the redness in a short- wavelength (violet) or a long-wavelength (orange) spec- tral light. a standard green was added to the spectral light. A nonreddish spectral light (i.e. one that is either greenish or uniquely yellow or blue) of course could not be cancelled by a green standard; if the light is greenish, then a red cancellation standard must be used. Jameson and Hurvich called the function which measures the intensity of the cancellation standard 1127
JAMES LARIMER. DAVID H. KRANTZ and CAROL M. CICEROSE
-.:o: 700 WAVELENGTH (nm)
Fig. 1. Opponent-cancellation coefficients for redness (open symbols) and greenness (filled symbols) for observers H (cir- cles) and J (triangles), for an equal-energy spectrum. Data replotted from Jameson and Hurvich (1955).The solid line is a linear functional for the CIE Standard Observer (Judd 1951).
that has to be added to each spectral light to cancel its greenness or redness the chromatic-response function of the red/green opponent process. Their measure- ments for two observers are shown in Fig. 1. In a simi- lar manner they measured the chromatic-response function of the yellow/blue opponent process, this time using standards that were yellow or blue (see Fig. 2). The solid curves in Figs. 1 and 2 are linear functions of the CIE tristimulus coordinates, proposed by Judd (1951) to describe the Hering theory.
400 500 600 700 WAVELENGTH (nm) Fig. 2. Yellow/blue cancellation coefficient (details as in Fig. 1).
When the redness in a violet light has been cancelled by adding green the endpoint of this procedure is a bluish--white light that is neither reddish nor greenish; we call this light a red/green equilibrium color. Cancelling the redness of an orange light pro- duces a yellowish white. which is also a red/green equi- librium color. Similarly. the cancellation endpoint obtained by cancelling the vellow in an orange light is a reddish color that is neither yellowish nor bluish. This is an example of a yellow/blue equilibrium color. A cancellation endpoint. or equilibrium color, is one that is either uniquely yellow. blue, or achromatic in the case of redjgreen cancellation; or uniquely red green. or achromatic in the case of yellow/blue cancel- lation. We denote the set of all red/green equilibria by A, and the set of all yellowblue equilibria by A?. Hurvich and Jameson implicitly assumed that each set of equilibria is closed under the linear operations of scalar multiplication and addition: (i) if a is in A,, then t * a is in Ai (i = 1,2); (ii) if a is in A,. then h is in Ai if and only if a 8 h is in Ai (i = I. 2). Krantz (1974) discussed these assumptions and showed that the two closure properties hold if and only if the corresponding chromatic-response function is a linear function of the tristimulus values based on metameric matching. More precisely. (i) and (ii) hold for Ai if and only if there exists a real-valued linear function 4, such that h is in Ai if and only if di(b) = 0. The function & is linear relative to color-mixture operations; that is. for any lights a, h and any s, t 2 0, 4i [(s * 0) o (t * b)] = Sh(a) + ttii(b). (^) (1) The function $i is consequently a linear function of any set of calorimetric primaries. It is measurable by the cancellation method; in fact, under these circum- stances, the chromatic-response function (4i as a func- tion of wavelength for an equalenergy spectrum) is in- dependent (except for a scale constant) of both the luminance level at which the measurement is made and the choice of the cancellation light. Another way to regard (i) and (ii) is in terms of the classical Bezold-Brucke and Abney hue shifts. Pro- perty (i) asserts that equilibrium colors remain so (and thus show no Bezold-Brucke hue shift) with changing luminance; while property (ii) asserts that equilibrium colors remain so under desaturation with other equi- librium colors. in particular, under desaturation with a true equilibrium white. Thus, the equilibrium colors exhibit no Abney hue shift. In particular, the yellowish and bluish equilibrium wavelengths must remain in red/green equilibrium when they are used to mutually cancel yellow/blue; therefore, a suitable mixture of them is achromatic. In short, opposite-hued equilibria are complementary. relative to a properly chosen equi- librium white. The cancellation experiment allows us to define new ways in which two lights can be equivalent. If two
1130 JAMES^ LARIMER.DAVIDH. KRANTZand CAROL.M. CICERON~
of this experiment. Relative luminance. for any fixed wave- length. was however very carefully controlled by careful filter calibrations. Subjects Two males and three females all with normal color vision served as observers in this experiment. Two of the female observers, PS and TC, were completely naive about the pur- pose of the experiment. Observers CC, DK. and JL were naive with respect to their own performance in the task. All observers used their right eyes. AIignment Subjects aligned themselves in the apparatus by adjusting the position of their bite bar while viewing a nearly white field of moderate intensity consisting of 650 nm light from M 1 or M2 and 505 nm linht from M2 or M 1. The alignment proceeded first by finding a head position in the beam which maximized the apparent brightness of the circle and which also yielded a first approximation to good focus of the stop ST,,,. Small adjustments^ were then made until^ no red or green fringes appeared at the edge of the circle. The M beam was not used in the experiments on red/green equili- bria; its alignment in yellowness/blueness experiments is de- scribed in part II of this series. Procedure After alignment observers were dark adapted for 10 min after which the experimental session began. Stimuli were exposed for 1 see with an intertrial interval of 20 set of dark- ness. There was no fixation point. so at the beginning of the I set exposure the observer had to make an eye movement to fixate the circle. Staircase procedure The sequence of stimulus presentations was determined by a staircase on the wavelength dimension. For example. to determine spectra1 equilibrium yellow, we might begin a sequence with a light of 565 nm. Generally, observers will call this wavelength greenish; a series of stimuli, say, 565,
- 575, 580nm would generate a sequence of responses greenish”, ‘*greenish”, “‘greenish”, “reddish”. respectively. At each change of response (e.g. from “greenish” to “red- dish”) the direction of the staircase is reversed. In one of the experiments to be described below a stair- cake was conducted using one monochromator while the other added a fixed-wavelength desaturant to the stimulus. In these experiments the desaturant is called the “addend” and the staircase or variable component to the mixture is called the “variable”. Different staircases are defined by dif- ferent luminance settings of the “variable” and “addend”.’ In order to reduce the possibility that an observer might anticipate the next stimulus (Cornsweet 1962; Nachmias and Steinman, 1965) several staircases were run simul- taneously and the experimenter would switch among them irregularly, doing only one, two or three trials on a single staircase at one time. Thus the sequence which the observer
2 Generally within a staircase brightness was maintained at a constant level with the exception that the luminance of the “variable” varied slightly with wavelength. This variabi- lity was due to imperfect com~nsation of small changes in the energy spectrum of the monochromators in any small region of the spectrum as well as differences in the lumino- sity functions among the observers. In all cases. these devia- tions were small.
saw might entail two trials on a low-luminan~ staircase to determine equilibrium yellow. followed by one trial on a high-luminance staircase to determine equilibrium blue, etc. All observers were uncertain what they would see next. Because of the moderate luminance and short stimulus exposures, the 2Osec intertrial interval was deemed sufll- cient to prevent adaptation effects across staircases, and it was verified that responses on a particular staircase at the very beginning of a session did not differ from those made after switching among staircases. Response The observer’s task was to judge whether the stimulus he had seen on a trial was reddish or greenish. Often this judge- ment was difficult since the stimulus might appear to be neither greenish nor reddish. The observer was instructed to make the “best“ guess possible, by responding to very minute hints of redness or greenness. Practice plays a significant role in an observer’s ability to respond to minute amounts ofredness or greenness. At first most observers perform reliably when the step size of the staircase is of the order of 5-1Onm. With practice all the observers in these experiments were able to perform consis- tently and reliably with a step size of 2-4 nm. Each observer had between 10 and 20 hr of practice under varying luminance conditions at this task before the data reported here were obtained. The nractice sessions also orovided use- ful initial estimates of the equilibrium loci. ‘
EXPERIMENT 1: SCALAR MULTIPLICATION TRAL RED/GREEN EQUILIBRIA
FOR SPEC-
Introduction A direct test of (i), closure under scalar multiplica- tion. was performed by determining the spectral loci (in the yellow and blue regions) of the red/green equili- bria, at several luminances. If closure obtains, then these spectral loci should be independent of luminance. These loci were determined using the staircase pro- cedure described previously. On a single day a deter- mination was made at all luminance levels. The exper- iment was repeated on each of 4 days. The exact spec- tral locus for a given luminance and hue on a single day was determined by linear inte~olation to the wavelength that would have generated 50per cent “too red” responses and 50 per cent “too green” responses. All staircases were obtained concurrently.
RfLSultS The four daily determinations at each luminance- hue combination were averaged. and an estimate of the standard error was computed using the between-day variability. A waveIength-by-lurni~n~ plot of these loci for each observer is given in Fig. 4. Note that the wavelength scale is expanded in the blue and yellow spectral regions. with a break in between. The horizon- tal bars at each point are the 80 per cent confidence in- tervals for the mean based on between-day estimates of the standard error. For linearity to obtain, the line connecting the various luminance levels at each unique hue should be vertical.
Opponent-process additivity-I 1131
+. 0 -.
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w. 5 +.5 -^ JL 5 o- (^12) -.5: &+ r E -1. :: 0 l 1.5 - (^) TC
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~%ikiF 5i5 ’ 5;5 ’ 5 WAVELENGTH (IIIII) Fig. 4. Mean value of equilibrium blue (left) and equilib- rium yellow (right) spectral loci, at various luminance levels, for five observers. Note the break in the wavelength abscissa between 485 and 565 nm. Error bars are 80 per cent confi- dence intervals based on between-session variability.
Only observers CC at yellow and DK at blue relia- bly3 differ from the linearity prediction. For all other observers and for CC at blue and DK at yellow, the spectral loci of the equilibrium hues, yellow and blue, are independent of luminance.
Discussion
Except for Cc’s equilibrium yellow and DK’s equi- librium blue, closure under scalar multiplication was strongly confirmed by this experiment. Had we chosen to plot the data on a 400-700 nm scale, then the devia- tions that did occur would appear trivial. And, in fact, they are visually trivial. The total shifts exhibited by CC and DK are both less than 5 nm, amounting to less than f 1 step on the staircases. Within-session and between-session criterion shifts of one step are fre- quently seen. If one considers that a 05 log unit incre- ment in luminance generates a marked increase in brightness, and also alters the saturation of blue or yel- low, it is quite possible that the shifts of CC and DK
3A significant variation from the prediction is opera- tionally defined to be any plot where the 80 per cent confi- dence intervals do not overlap. From a purely statistical standpoint. the error term should be based on days x lu- minance interaction. but we felt that the between-days error was far more meaningful in the present context.
are due to redness/greenness criterion shifts associated with these changes in brightness and saturation. Furthermore, there is no indication of a trend that holds up across the five observers. (In part II of this series we show that a relatively small yellow/blue non- linearity is detectable by our methods. since every observer shows the same trend.) The luminance range spanned in this test covers a substantial part of the most interesting range for color vision. There would not be too much point. at this stage, in pursuing dimmer yellows. which would appear nearly white. or brighter blues (the latter would have been hard to obtain with our apparatus). More- over, a factor of 10 to 100 is a very significant range for any system to operate linearly. There is no question that the individual differences between observers, particularly in the yellow, are reli- able. They must be caused either by differences in pho- topigment absorption functions or differences in the coefficients of the photopigments in the linear function 4,.
EXPERIMENT 2: ADDITION OF RED;GREEN EQL’ILIRRIA Introduction A direct test of condition (ii) would be to take the equilibrium yellow and the equilibrium blue wave- lengths and mix them in some luminance ratio. If the mixture were judged neither reddish nor greenish, then (ii) would be confirmed. It is clear from Fig. 4, however, that the concept of “equilibrium wavelength” is a statistical one, subject to moderate day-to-day variability. It is not surprising to find a shift of, say, 3 nm between days; in which case the wavelength that is the 50 per cent point on one day may be judged “reddish” or “greenish.” with perhaps 90 per cent consistency, on another day. These fluctua- tions are most likely due to criterion shifts; as dis- cussed below. the stable criterion is a product of con- siderable practice. Therefore, it is far more reasonable to test (ii) by fixing one wavelength component (called the addend) and then using the staircase method on the other component (the variable). This method deter- mines an equilibrium mixture with the same expe.r- imental design and the same statistical properties that characterized the determinations of single equilibrium wavelengths shown in Fig. 4. The test, then, would be to allow the addend to be (say) a blue equilibrium wavelength and to see whether the equilibrium yellow wavelength is the same with or without the blue addend and independent of the intensities of the blue and yellow components. A further refinement of this idea was used, however, for three reasons. First, the above test requires prior determination of the equilibrium wavelength to use as addends. This requires temporal separation of the two experiments, with consequent possible criterion varia- tion. Secondly, the above method would make the ex- periment particularly vulnerable to lack of statistical
Opponent-process additivity-I
460- : : ,
450-'& 550 560 570 580 590 600 YELLOWISH COMPONENT WAVELENGTH (nm) Fig. S(a)
3 A I : 450550 560 570 ; 560 590 600 YELLOWISH COhPGNFNT WAVELENGTH (nm) ‘5’ Fig. 5(b)
48 500/- OBSERVER DK $I.30 Y (^) EQUAL : : P 0.
/
0, : : I
550 560 570 580 590 800 YELLOWISH COMPONENT WAVELENGTH (nm) Fig. 5(c)
5OOr
5 460 2 t 450' I 550 560 570 560 590 600 YELLOWISH COMPONENT WAVELENGTH (MI) Fig. 5(d)
550 560 570 560 590 $ YELLOWISH COMPONENT WAVELENGTH (nm) Fig. 5(e)
Fig. 5. Loci of yellow-plus-blue mixtures that are neither reddish nor greenish. Shape of symbol denotes approximate luminance ratio of yellow to blue [log,, (luminance Y/ht- minance B)] : X-l .40; 0-0~90-1~10; &-@4Z-O~50; G--000; A--0.45. Position of bar on symbol denotes ap- proximate retinal illumination from yellow component. Upper bar: high luminance; middle bar: medium luminance; lower bar: low luminance. (Respectively 324, 115 or 41 td.) Straight lines were fit by eye to each set of points having a fixed luminance ratio. Poorly determined points are indicated by parentheses. Heavy solid lines are more reliably determined than light dashed ones. The monochromatic equilibrium wavelengths (average from Fig. 4, for each observer) are the ordinate and abscissa of the point indicated by the large star. Under linearity, we predict that the locus for a given luminance ratio is independent of the luminance of the yellow component; that the different loci intersect at the star: and that the ratios of their slopes at the star are equal to the ratios of the corresponding luminance ratios.
1 I.3 JA,MESLARISER. DAVID H. KKANTZ. and^ CAROL^ M.^ CII~WII~L
deterlnination of his spectral equilibr~ obtained in ex- periment I. Straight lines were fit by eye to each set of points having a fixed luminance ratto. The expected curv!e [eyttationf2)] seems to be approximated welt enough ‘ox a straight line over the relatively small wavelength ranges involved. For two of the log ratios, 1,40(X) and -045 (A). either the yellowish component (X) or the bluish component (a) was several times more intense than the other component. In these mixtures the less intense component slightly desaturated the mixture. but the appearance of the mixture would be described subjectively aseither very yellow or very blue. At these ratios a slight shift away from the locus of the unique hue in the more intense component of the mixture caused a dramatic shift in the dim component. The extreme points on these curves were subject to con- siderable error. Some observers IFS. CC. JL and TC) were not able in every session to adjust the dimmer variable to cancel the redness or greenness in the more intense addend. and these mixtures are indicated by placing parentheses around the data point. We have drawn those tines as dashed to indicate that they are not as well determined as the sohd lines which can be considered good linear approximations to the actual curves near the equilibrium hues. The plots in Fig, 5 strongly confirm the expectation that mixtures in a fixed ratio lie on a single-valued curve and that these curves intersect at the locus of the equilibrium wavelengths. thus confirming property (ii). Property (i) implies that a change of overall luminano? of the mixture should have no effect on the wavr- Iength x wavelength Iocus of the data point. This pre- diction is also strongly confirmed. and for some observers (i.e. PS, DK. JL and TC) luminance differ- ences of as much as 0.9 Jog,, units of otherwise identi- cal mixtures yielded virtually congruent data points. Again. the three observers who deviate from the pre- diction do so in no systematic Lshion and by amounts ‘that are visually trivial. Finally, in Fig. 6, we plot the logarithm of the slope. dpjdi,, against the logarithm of the luminance ratio. SY. for all 19 lines (5 observers of Fig. 5). From equation
- we expect that log(dp/d/.) = log (s/r) - log/&ii.)’ 4‘r(rt)l. The predicted 45” line fits weli, and the esti- mate value of log/ ~~~~)~~~~~~)~varies little across observers. The line drawn in Fig. 6 corresponds to &~‘,(;.)/&b) = - 1.9, which is not far from the value that would be estimated from the ratio of the slopes of #, araund 580 nm and 475 nm in Fig. I.
The results of experiment 2 strongly confirm the additivity property and reconfirm the results of exper- iment 1. When addends were chosen that were equili- bria. the correspondiu~ variable was always within I or 2 nm of the appropriate equilibrium wavelength determined in experiment 1. and this difference was always well within the between-day ~driability mea- sured in experiment t (e.g. see observer JLs data
1.2; (^) t- PS L. .sl
l cc
1
3 DK l .Tt 3 TC /
!! :
-.6 -.4 0 4 .@ 1. ‘qra(s/t) Fig. 4. The logarithm of the slope (dp/di.) of each line in Fig. 5 is plotted against the logarithm of the luminance ratio (s,t = yellow’blue) that generated the line. From equation 3 the prediction is a 4.5’ straight line with intercept -logl~‘,(i)!@,(~)l. The least-squares 45’ line is shown.
points (578. 478) for “equat” and -0.45 luminan~ ratios).
GENERAL DtSCI_‘SSfD1V: CONDITIONS FOR ADDIT~~IT~ Under the conditions of our experiments. additivity holds for red/green equilibria. Under what circum- stances wit1 it fail? A compbte answer to this question constitutes a broad program of research. Nevertheless. we feel that it is necessary to mention some of the boundary conditions that may be important, at least in order to facilitate others’ repeating our results.
We used a moderately large (26 f centrally fixated field, with exposures of 1 sec. Some clue as to how field size, retinal locus, and exposure duration bear on the results may be obtained from previous studies. For example. Ingling. Scheibner and Boynton (1970) and Snvoie (1973) failed to obtain results consistent with ours. in studies that involved a small (3’) or a brief (5 msecf test light, respectively. However. the logic of their methods was quite different also {see next section) and that may be as important as the test-field para- meters.
These have enormous and fairly weii understo~ effects on color appearance, and hence on the spectral loci of the equilibria. A light which is achromatic or yellow in neutral conditions will appear greenish if the eye is red-adapted or if a reddish surround is present. Not only are the sets of equilibrium and achromatic lights different as a function ofadaptation or surround, but possibly their additivity properties will change. For example, if the equilibrium yellow shifts to Longer
1136 JAMESLARIMER.DAVIDH. KRANTZ and Cabot. M. CicEaoNt
574.5 nm and varied only slightly and non-systemati- cally over a 1.3 log unit change in intensity. His white desaturant also had no effect. The range of values obtained in his sample was very similar to ours. (We do not know the exact parameters of his test light; his observers were daylight adapted.) Our experiment differs from his chiefly in the use of a more controlled psychophysical method and in its much greater variety of addition experiments (Experiment 2). That the results coincide is reassuring; the only discordant note is that he also measured “fundamental” green and red loci, and his results and conclusions for those deter- minations are quite different from ours (Larimer. Krantz and Cicerone, 1974). Purdy (193 1) repeated Westphal’s ex~riment on his own eye, obtaining “fundamental loci of 476 and 576nn1, with only slight variations at 10, 100 and loo0 td. He also made some observations with admix- ture of a “white” desaturant; he asserted that the hue of 580~1 is unchanged but that 480 and 470 nm become pinkish when white is added. Boynton and Gordon (1965) used a color-naming method to determine equilibrium loci. Their equilib- rium yellow and blue were defined by equality of “red” and “green” color-name scores. Since their color-name scores take salience of a hue component into account ( f,2 or 3 points are possible per trial). one would expect their “equilibrium blue” to occur at shorter wave- lengths than those obtained by forced-choice methods and also to shift to even shorter wavelengths at high luminance. This prediction is based on the asymmetry in the salience of redness vs greenness in the 46O- 485 nm range, due to veiling of redness by blueness (see discussion of perceptual learning. supra). That is what happened, for all three of their observers: their IOO-td blue equilibria were 465, 474 and 470 nm. while their lOOO-tdones were 463, 462 and 463 nm. respectively. This result could conceivably be due to visual para- meters (e.g. they use @3 set flashes and a fixation point) but very likely it is due to the nature of the color-name score as just discussed. For equilibrium yellow. their observers fell within the usual range. and shifts were nonsystematic (575.573 and 584 nm at 100 td: 573. 584 and 584 nm at 1000 td). Our results can be seen as confirming the main con- clusionsof Westphal and Purdy, with a more elaborate multiple-staircase method and perhaps with more pre- cisely specified conditions. We extend their results to a wider range of additive mixtures; and we find that. over such an extended range, the linearity hypothesis yields truly excellent predictions.
To make clear the relation between our work and the work on hue shifts, we need to introduce an ad- ditional bit of formalism. We have already used A, to denote the set of all red/green equilibrium colors and A2 for yellow/blue equilibria. Let C, denote the set of lights that are constant in hue when radiance changes.
and let C, denote the set of fights that are constant in hue when a particular desaturating light w is added. In other words,
uisinC,ifandonlyifa-n,,r*a.forallt>O; 0 is in C, if and only if a _ uII a 0 (r * WLfor all r > 0.
The sets C, and C,. are sets of invariant lights. respect- ively for the Bezold-Briicke effect and the Abney effect (relative to n’ as desaturant). Two points should be noted about these newly defined sets. First, the definition depends on an empiri- cal relation of hue matching, denoted here by -hua. This is a crucial element for any interpretations. Secondly, C, or C, may very we11be empty. ?Jothing excludes this possibility. But A, and AI are surely not empty; we were bound. in our studies, to identify lights which were equilibrium lights. To relate the sets Ai to C,. we need to answer three interrelated questions: (a) Is C, nonempty? That is, are there invariant hues with respect to the Bezold-Brticke shift? (b) Is Ai radiance-invariant, that is, does hypothesis (i) hold? (c) Is Ai part of C,? The logical interrelations among questions (at(c) are as follows. Question (c) presupposes an affirmative answer to (a). If (a) and (c) are both answered posi- tively. then obviously (b) is also affirmative. What is slightly less obvious is the converse: if(b) is answered affirmatively, then so are (a) and (c). The reason is found in the definition of hue match. We use the term “hue” in such a way that any two yellows that are neither reddish nor greenish match in hue. That is. any two yellow colors in A, are a hue match, and likewise for any two blues or any two whites. Therefore, if A, satisfies hypothesis (i), then any yellow light in A, that remains yellow (rather than going white or blue) with changes in radiance is in C, and likewise for any blue in Ai that remains blue as radiance changes. A similar argument applies if A2 satisfies (i). If both AI and A, satisfy (i), then every equilibrium color, including white. is hue-invariant, so A, and A, then are entirely contained in C,. Naturally. it is logically possible that there are invar- iant colors that are not equilibrium colors, whether or not hypothesis (i) holds. So it is possible that only (a) would be answered affirmatively. Previous studies of the Bezold-Briicke effect yield answers to question (a) and only secondarily to ques- tion (c). Purdy (193 I) found three monochromatic C, lights (invariant wavelengths) at 474, 506 and 571 nm. as well as a mixture of long and short-wavelength light that was a C, light (invariant bluish-red). He therefore claimed an a~rmative answer to (a). He found that the three monochromatic C, tights were nearly the same as “funda~n~~ biue, green and yellow, but that the red C, light was bluish red, rather than “fundamental” red. For the A, equilibria, therefore, he concluded that (c) is affirmative.
Opponent-process additivity-I 1137
Jameson and Hurvich (1951) made a series of invar- iant wavelength determinations in different adaptation states. For neutral adaptation, blue, green and yellow invariant points were located confirming Purdy’s result. For two observers, all three invariant points fall well within the distribution of our measurements of equilibrium wavelengths: 476, 499, 580 nm and 478, 494,582 nm. The third observer yielded a more deviant result: 466, 491 and 587 nm. They did not test (c) di- rectly, but the close correspondence to our equilibrium determinations for two observers is suggestive of an affirmative answer to (c). Boynton and Gordon (1965) used three different cri- teria for hue matching: simultaneous matching for steady lights, simultaneous matching for 0.3 set flashes, and a derived “match” based on color naming of 0.3 set flashes. They found bluish, greenish and yellow- ish invariant wavelengths by all three methods, but the locations varied considerably with the method, and for no method did all three of their subjects produce invar- iant points near the equilibrium points. The closest correspondence was for the steady lights, in which Purdy’s method was replicated and his results were generally well reproduced. Savoie (1973) cast doubt on the existence of invar- iant hues, through his failure to find one in the yellow region of the spectrum, using a staircase hue-matching technique, with 5 msec flashes. It is possible that this assortment of results corres- ponds to the variety of viewing conditions employed (bipartite vs homogeneous fields, fixation targets, flashes vs steady viewing etc.). It is also possible that some of the “invariant” points are not truly invariant. Savoie found nonmonotonic variation of hue with radiance for a constant wavelength. and pointed out that given such nonmonotonicities, it is possible to find spurious “invariant” points if only two intensity levels are used (as was the case in the Hurvich and Jameson and Boynton and Gordon studies). Our results, however. show that the yellow and the blue equilibrium wavelengths are, to an excellent approximation, radiance invariant. We set out to answer (b), rather than (a) as in the bulk of previous work; but our affirmative answer to (b) implies, as noted above, that (a) and (c) are also to be answered affirmatively, at least in so far as the yellowish and bluish red/green equilibria are concerned. Our results thus confirm Purdy’s conclusions. though our initial goal, method, and viewing conditions were all quite different from his. There is no question that both hue matching and color naming give very valuable information about the general features of the Bezold-Briicke shift. For all three of Boynton and Gordon’s methods, and for all their subjects, the short-wavelength end of the spec- trum became relatively bluer, compared to red, and the long-wavelength end became relatively yellower, com- pared to red. as luminance increased. The same results were obtained by Purdy; and indeed, this general qua- litative finding can be quickly verified by anyone with
access to a monochromator and a 1 log,,, unit density filter. Heteroluminous hue matching is an extremely diffi- cult task, as Purdy noted, and it is possible that certain quantitative results obtained by this method have sys- tematic errors. It is not clear, in general. how to iden- tify “error”. One principle. which we favor strongly. is the adoption of a definition of “hue match” such that any two equilibrium yellows are a hue match. and like- wise for any two equilibrium greens and any two equi- librium blues. [Please note that adopting this principle does not force the result that (b) is answered affirmati- vely, since it would still be possible for two equilibria at different luminance levels to have different chroma- ticities.] The Boynton and Gordon hue-matching procedures do not always satisfy this criterion. As we previously noted, one of their subjects had an equilibrium yellow of 584 nm at 1000 td and 573 nm at 100 td. But his hue- match (for the same 0.3~set flashes) was between 584 nm at 1000 td and 587 at 100 td. The defined “hue match” (equal ratios of adjusted color-name scores) was in this case even more deviant: 584 run. 1000 td, to 595nm at 100 td. Their data show other instances which, while less severe. are nevertheless statistically and in some cases visually significant. Savoie’s hue matches cannot be subjected to this kind of internal analysis, since he did not determine a yellow equilibrium wavelength. The observer (princi- pally Savoie himself) was required to judge only whether the comparison wavelength was “redder” or ‘greener” than the standard, with nothing at all said about “yellow”. The consequences for hue matching of entirely ignoring yellowness as a perceptual quality are unpredictable. We tried out Savoie’s response mode in a successive matching technique, requiring the observer to judge whether the comparison stimulus was redder or greener than a standard equilibrium yellow. It was easy to verify that the 50 per cent “redder” point was invariant with luminance. In sum we observe that focusing on question (a), that of invariance of hue. requires hue matching, either directly. or indirectly by processing of other color re- sponses. These methods are difficult, and in some in- stances they fail a test which we regard as a basic check on validity: like-colored equilibria should match in hue. The results of hue matching studies are equivocal with regard to the existence and the location of invar- iant hues. We, on the other hand. focus on question (b) and thus find answers to (a) and (c). On the other hand, our method has no way to locate nonequilibrium in- variant hues, if there are any. nor does it assess the di- rection or the magnitude of hue shifts for noninvariant colors. The relation between the equilibrium sets Ai and the set of C, (no Abney effect) lights can be analyzed simi- larly into questions (a’)-@‘): (a’) Is C, nonempty: That is. are there invariant hues with respect to desaturation by w?
Opponent-process additivity-I 1139
There are many other possibilities. Nevertheless, a very powerful constraint has been obtained, in that many possibilities have been eliminated.
sponses and spectral saturation. J. opt. Sot. Am. 45, 546
REFERENCES
Jameson D. and Hurvich L. M. (1967) Fixation-light bias: an unwanted by-product of fixation control. Yision Res. 7,805-809. Boynton R. M. and Gordon J. (1965) Bezold-Briicke hue Judd D. B. (1951) Basic correlates of the visual stimulus. In shift measured by color-naming technique. J. opt. Sot. (^) Har~dhook ofE.sperirrwta/ P.s~cholog!~ (Edited by Stevens Am. JS,78-86. S. S.). Wiley. New York. Cornsweet T. (1962). The staircase method in psychophy- sics. Am. J. Psycho/. 75, 48%491. Grassmann H. (18534) Zur Theorie der Farbenmischung. Poggendorfi Ann. d. Physik (Leipzig) 1853, 89, 6%84. (Engl. Transl. in Phil. Msg. 1854. Ser. 4. 7. 254-264). Hering E. (1878) Zur Lehre vom Lichtsinne. C. Gerald’s Sohn. Vienna.
Krantz D. H. (1974) Color measurement and color theory -II: Opponent-colors theory. J. Math. Psychol. I1 (in press).
Ingling C. R., Jr., Scheibner H. M. 0. and Boynton R. M. (1970) Color naming of small fovea1 fields. Vision Res. 10, 501-511. Jameson D. and Hurvich L. M. (1951) Use of spectral hue- invariant loci for the specification of white stimuli. J. exp. Psychol. 41,455-463.
Larimer J.. Krantz D. H. and Cicerone C. M. (1974) Opponent-process additivity-II: Yellow!blue equilibria and nonlinear models c’isio!l Rrs. (to be submitted). Nachmias J. and Steinman R. M. (1965) An experimental comparison of the method of limits and the double stair- case-method. Am. J. Psgchol. 78, I 12-l 15. Purdy D. M. (1931) Spectral hue as a function of intensity. Am. J. Psychol. 43, 541-559.
Jameson D. and Hurvich L. M. (1955) Some quantitative aspects of an opponent-colors theory-I: Chromatic re-
Savoie R. E. (1973) Bezold-Briicke effect and visual non- linearity. J. opt. Sot. Am. 63, 12531261. Westphal H. (1909) Unmittelbare Bestimmung der Urfarben. Z. Sinnesphysiologie 44, 182-230.
R&m&--Une lumiitre d’equilibre rouge/vert n’apparait ni rouge ni verte (c’est g dire uniquement jaune. uniquement bleue ou achromatique). On dktermine g divers niveaux de luminance un ensemble d’equi- libres rouge/vert spectraux et non spectraux, afin de savoir si l’ensemble de tels irquilibres est ferme dans des operations lineaires de melanges de couleurs. Les lieux spectraux d’equilibre jaune et bleu ne montrent pas de variation. ou des variations visuelle- ment insignifiantes, dans une marge de l-2 log,,, unit&. 11n’y a pas de tendance ripktable pour les obser- vateurs. On dtduit que les equilibres spectraux rouge/vert sent fermCs vis-&vis de la multiplication sca- laire; ce sent done des tonalit& invariantes vis-&is de l’effet Bezold-Briicke. Le mtlange additif des longueurs d’onde d’equilibre jaune et bleu. g tout rapport de luminance, est aussi une lumikre d’kquilibre. De petits changements de la composante jaungtre du melange vers le rouge ou le vert doivent etre compensis par des changements prCvisibles de la composante bleuitre du mklange vers le vert ou le rouge. On con&t que les iquilibres jaune et bleu sent complt?mentaires vis-8-vis d’un blanc d’equilibre; que la dksaturation d’un iquilibre jaune ou bleu avec un tel blanc ne produit pas de decalage de tonalitt d’Abney; et que l’ensemble des equilibres rouge/vert est fermi vis-h-vis des operations lintaires gin&ales. En contiquence la fonction chromatique de riponse rouge/vert. mesurte par la technique de Jameson- Hurvich d’annulation g l’iquilibre, est une fonction lineaire des coordonnies de melanges de couleurs du sujet. Une seconde contiquence de la fermeture liniaire de l’equilibre est une forte contrainte sur la classe des regles de combinaison par lesquelles les reponses des recepteurs sent recodees dans le processus anta- goniste rouge/vert.
Zusammenfassung-Ein gleichanteiliges rot-griines Licht ist ein solches. das weder rBtlich noch griinlich erscheint (d.h. entweder einheitlich gelb. einheitlich blau oder unbunt). Eine Reihe spaktraler und nichts- pektraler Rot-Griin-Gleichgewichte wurde mr mehrere Leuchtdichteniveaus bestimmt. urn zu unter- suchen. ob die Gesamtzahl aller dieser Gleichgewichte durch lineare Farbmischoperationen bestimmt ist. Die Farborte des gleichanteiligen Gelbs bzw. Blaus zeigten entweder iiberhaupt keine oder nur eine visuell nichtsignifikante Variation iiber einen Bereich von l-2 logarithm&hen Einheiten. Unter allen Versuchs- personen gab es keinerlei reproduzierbare Tendenzen. Wir schlossen. dass spektrale Rot-Griin-Gleichgew- ichte durch eine skalare Multiplikation bestimmt sind: folglich sind sic inwriantc Farhtiinc hcriiglich der Bezold-Briicke-Verschiebung. Die additive Mischung gelber und blauer Gleichgewichtswellenllngen ergibt in jedem Leuchtdichte- verhlltnis ebenfalls ein gleichanteiliges Licht. Kleine Verlnderungen..der Gelb-Komponente einer Mis- chung in Richtung Rot oder Griin miissen durch vorherbestimmbare Anderungen der Blau-Komponente der Mischung in Richtung Griin oder Rot kompensiert werden. Wir schlossen. dass gelbe und blaue Gleichgewichte in Bezug aufein gleichanteiliges Weiss komplementlr sind; die Verringerung der Farbtit- tigung eines gelben oder blauen gleichanteiligen Lichtes durch Beimischung eines derartigen Weiss verur- sacht keine Abneysche Farbverschiebung; ausserdem folgerten wir, dass die Reihe der Rot-Griin-Gleich- gewichte durch allgemeine lineare Operationen bestimmt wird.
1140 JAMI.SLARIMI:R.DAVII) H. KRASTZ^ and^ CAROL^ M.^ ~ICI:ROUI
Eine Konsequenz ist. dass die mit der Jameson-Hurvlch-Technih des Gleichgewlchtsabgleichs gemes- scne rot-giine Farbiibertragungsl’unktion eine linrare Funktion der Farbvergleichskoordinaten der Ver- suchsperson ist. Eine zweite Konsequcnz der linearen Gesetzmlssigkeiten ist eine starke EinschrPnkung fir die Klassc der Kombinationsregeln. nach denen die Ausgangssignale der Rezeptoren fiir dicsen Misch- prozess kodiert werden.
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