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High-level trichromatic color matching and the pigment-bleaching hypothesis
HIGH-LEVEL TRICHROMATIC COLOR MATCHING AND THE PIGMENT-BLEACHING HYPOTHESIS GUNTERWYSZECKI National Research Council of Canada. Ottawa KIA 0R6, Canada and
W. S. STILES 89 Richmond Hill Court. Richmond. Surrey. England (Receiced 20 April 1979) color matches with reference to a given “white” stimulus at a retinal illuminante of 100,000 td can be predicted from Maxwell-type color matches made with reference to the same “white” stimulus but at a retinal illuminance of loo0 td by means of a simple hypothesis of bleaching of the visual pigments. The spectral distributions of the fundamentals of color vision. the densities of the visual pigments of the 1000 and 100,000 td levels, as well as the spectral transmittances of the lens and macula pigments which must be assumed as essential parameters of the visual mechanism emerge with values which appear consistent with those obtained from other investigations. While both the “red” and the “green” fundamentals exhibit strong and predictable bleaching characteristics, the “blue” fundamental appears to bleach only very little under the given experimental conditions. The somewhat unexpected behavior of the “blue” fundamental is not explained, but it is suggested that the pigment associated with the “blue” fundamental may be in a near-diluted state already at moderate levels of retinal illuminance, making it difficult to detect bleaching at higher levels. Abstract-Maxwell-type
the B primary so that the mixture [L(/.)L + B(i)B] color matches the mixture [R(i.)R- + G(i)G]. When the amount of the desaturating B primary is very small. it is sometimes helpful to add a small amount of the B primary to the upper half of the bipartite field and then take as the match the difference in the amounts of the B primaries in the two halves. The amounts R(i). G(i). B(L) of t,he three primaries required in the match are called the tristimulus values of the test stimulus L(i.)L. The amount of the desaturating primary is given a negative sign. Apart from differences at test stimuli of short wavelengths consistent with different absorptions in the eye lens and macular pigment, the respective color matches of normal trichromatic observers agree rather closely, leading to similar tristimulus values R(i), G(i) and B(i) for a given stimulus L(i)L. From such data the familiar color-matching functions t+.), a(;.), 6(i) emerge:
INTRODUCTION Color matching is one of the most fundamental psychophysical experiments in color vision. Color-matching data provide the basis for color vision theories as well as the basis for calorimetry. There are two main methods that have been used in color-matching experiments: the maximum-satu:ation method and the Maxwell method. Figure 1 gives the parameters which are essential to the two methods. A bipartite visual field is usually used in either method. In the illustrated case the field has a horizontal dividing line. There are three fixed primary stimuli R, C, B which are normally monochromatic stimuli consisting of narrow bands of wavelength i, one in the red, one in the green and one in the blue part of the spectrum. Then there is a variable test stimulus L, normally also monochromatic but of variable wavelength i., and finally a fixed reference stimulus w, normally of a “white” color. The maximum-saturation method shown on the left of Fig. 1 is the most commonly used method. The test stimulus L, of magnitude L(L). presented in the lower half of the bipartite field, is “desaturated” by the minimum amount required of one of the three primary stimuli R, G, B such that the mixture of the test stimulus and the desaturating primary can be color matched by an appropriate mixture of the remaining two primaries appearing in the upper half of the bipartite field. The choice of the desaturating primary depends solely on the wavelength of the test stimulus relative to the wavelengths of the primaries. In the case illustrated, the test stimulus L is desaturated by
F(i) = R(i.)/L(i.) g(i) = G(i.)jL(i.) 6(i) = B(i.)/L(i)
(1)
The Maxwell method shown on the right of Fig. 1 employs a fixed reference stimulus m, normally of a “white” color, appearing in the upper half of the bipartite field. In the lower half, a mixture of the test stimulus L and two of the available three primary stimuli (R, G and B) is presented. which when appropriate amounts of them a& chosen, color matches the reference stimulus. The wavelength of the test stimulus determines which two of the three primaries are 23
G. %‘YSzfCKl snd ‘A’. s.
Fig. i. Color matching in a bipartite visual field using the maximum-saturation me&hod or the lMaxwell method. R, G. B = Primary stimuli (of fixed wavelengths i,. i.,. i.,) R(b). G(i). S(L) = Amounts of primary stimufi (tristimuhts values) (e.g. in terms of radiance: W m - z sr - ’ ) g = Variable test stimulus of wavelength E. L(L) = Amount of variable test stimulus (e.g. in terms of radiance: W rneL St--‘) @ = Fixed reference stimulus (e.g. of “white” color) W = Amount of fixed reference stimulus (e.g. in luminance or trolands)
added to the test stimulus. In the example illustrated, the stimulus WV color matches the mixture [R(EJR + G(L)c + t(i.)L]. As in the maximum-saturation method, normal trichromatic observers agree
rather closeiy with their respective color matches, leading to similar tristimulus values. The results of a complete color-matching experiment by means of the Maxwell method can be illustrated graphically as shown in Fig. 2. From this graph one can read off the tristimulus values t(i), R(i.), G(i) and B(E.)of each test stimulus L of wavelength i. or wavenumber nr in terms of the radiant power manowatts, nW) of the monochromatic stimuli involved. For example. when the test stimulus L is set at m = 16,500 cm- ’ (or i 2 606 nm), a mixture of L, G and B using amounts L(L) = 0.399, p;iL) = 0.355 and Bfi.) = 0.344 nW, respectivefy. are required by the observer to obtain a color match with the reference stimulus w at 1000 td. At wavenumbers m = 17,6OOcm-’ and m s 20,3OOcm-’ we notice certain singularities in the Maxwell spectral distribution curves of Fig. 2. When the test stimutus L is set at these wavenumbers only one, instead of two, of the three primary stimuli is required to provide. when mixed with .E, a color match with @. With L at 17,6GQcm- ’ the addition of B suffices, and with L at 20,300 cm- * the addition of I? suffices. These results are quite expected and in accordance with the laws of trichromatic color matching.
WAVELENGTH 450 1
400
STILES
X (s-n)
500 i
5.50
600
I
I
650
700 I
,iiX)
L(X)
~
L
‘-‘25
24
22
23
ZI
20
B WAVENUMBER
19 G m
I8
17
16
15
14103
ii
(cm*‘)
Fig 2. Maxwell spectral distribution functions L.(i.).R(i.). G(i.) and B(L) obtained by observer CW in a 2” bipartite visual field. The reference “white” stimulus is of constant radiance providing a retinal illuming ance of loo0 td (1 ktd) and consists of a mixture of the three instrumental primaries R. 6 and B of radiant powers equal to 1.0856. 0.5567 and 0.3515nW. respectively.
25
High-level trichromatic color matching
The data shown in Fig. 2 were obtained by observer GW (then of age 51) on NRC’s trichrornator. The observing conditions and instrumental parameters were similar to those described in the pilot study published earlier (Wyszecki, 1978): Visuuffiefd:
2’ bipartite circular field with dark sur-
round, seen in Maxwellian view. The entrance slits of the three monochromators which make up the trichromator are imaged in the center of the natural pupil. These images are small enough (not more than 2 mm dia in any direction) to be unobstructed by the eye’s iris diaphragm. Instrumental primaries (R, G, B): f? centered at wavenumber ma = 15,5OOcm-’ (z 645 nm) G centered at wavenumber mG = 19,000 cm- l ( 5 526nm) II centered at wavenumber ma = 22,5OOcm-’ (z 444nm) The width Am of the waveband of each primary stimulus was the same and set at 500 cm- I. Test stimuli (L): L centered at wavenumbers
rangfrom m = 14,500 cm- ’ (369Onm) to m = 24,5OOcm-’ (t 408 nm) at regular intervals of 250 cm-i. The width Am of the waveband varied with the wavenumber m of the test stimulus. However, in using three different slits when traversing the long-, middle- and short-wavelength regions of the spectrum, it was possible to have wavebands Am = 5OOcm-’ when m was equal to mR, mc or ma. Thus, whenever the test stimulus L was centered at mR = 15,5oOcm-‘, mG = 19,OOOcm- or ma = 22,500 cm- ‘, an appropriate radiance adjustment was all that was needed to obtain a complete physical match between L and one of the primary stimuli, R. G or B. Reference stimulus (w): Stimulus of average daylight color with (x, y)-chromaticity of CIE standard illuminant D6s. The reference stimulus was produced by an appropriate mixture of the instrumental primaries l?, G and 8. To obtain a retinal illuminance of 1000 td, the required radiant powers for the three primaries were ing
Rw = 1.0856nW
Gw = 0.5567 nW Bw = 0.3516 nW The fact that w is an additive mixture of the three instrumental primaries anchors the Maxwell spectral distribution curves of Fig. 2 at the wavenumbers of the primaries l?, G and B as indicated by solid circles. That is, whenever the test stimulus L has a wavenumber identical’to that of a primary stimulus, both the reference stimulus w, presented in the upper half of the bipartite visual field, and the additive mixture of stimuli presented in the lower haif of the field become physically identical as soon as a color match is produced by the observer.
The results of the color-matching experiment by means of the Maxwell method as illustrated in Fig. 2 can be converted to the familiar color-matching functions i(;.), d(;.b 6(i.). The following equations are used: i(j) = g(i.) =
&.) =
Rw - R(i) L(j.) Gw - G(L)
(2)
L(i) Bw - i?(i.) L(i)
I
As can be seen from Fig. 2, L(i) never becomes zero, whereas R(i.), G(1) and B(1) are respectively zero in the long-, middle- and short-wavelength region of the spectrum. The quantities Rw, Gw and Bw are fixed throughout the experiment and define the color of the reference stimulus w. It is clear that if the basic laws of trichromatic color matching, particularly the proportionality and additivity laws, hold strictly, the color-matching functions ?(i.), g(1) and 6(i) derived from Maxwell matches in accordance with equations (2) must (within the experimental uncertainties) be identical to those derived more directly by means of the maximum-saturation method (equations 1). However, there is experimental evidence (for example, Crawford, 1965; Lozano and Palmer, 1968) of systematic differences between the two sets of color-matching functions obtained by the two different methods. In this study we will assume that color matching done at a given level of retinal illuminance and with a fixed relative spectral composition of the “white” reference stimulus ‘strictly obeys the color-matching laws. With these restrictions the laws reduce to asserting a more limited additivity law, namely, that if any two test stimulus mixtures A and B match individually the fixed reference stimulus, then so does any additive mixture of A and B in amounts pA and (1 - p) B respectively where p is less than unity. Observations by Lozano and Palmer (1968) using a 10’ matching field suggest that even this limited additivity law may not be strictly obeyed. But a critical study for a 2” field has not yet been made and in what follows the validity of the limited additivity law will be assumed. Our attention will be directed to a failure in the proportionality law of color matching when the level of retinal illuminance is changed from -a moderate level (1000 td) to a high level (100,000 td), the relative spectral composition of the reference stimulus being kept the same. As described in the pilot study (Wyszecki, 1978), some color matches made at 1000 td clearly do not hold when viewed at 100,000 td. In fact, the breakdown of the proportionality law was observed to set in at approx 8000 td and appeared to be completed at approx 50,000 td. Above 50,000 td the proportionality
26
G. WYSZECKI and W. S. STILES
law appeared to hold again. For some color matches. the magnitude of the breakdown can be quite large when the 50,OtMltd level is reached. It is generally assumed that the breakdowns of the color matches caused by increases in the retinal illuminance can be attributed to bleaching of the photopigments, particularly the pigment (or pigments) assumed to reside in the “red-receptor mechanism” (Wright, 1936; Brindley. 1953; Terstiege, 1967: Ingling. 1969). In a thorough study just published by Alpern (1979), it is shown that the fraction of the red stimulus in a red-green mixture of stimuli. colormatched to a yellow stimulus. increases when the level of stimulation, in a Maxwell-type experiment. is sufficiently increased to bleach the “red” (erythrolabe) and the “green” (chlorolabe) pigments. The changes in color matching in the yellow region of the spectrum, observed by a normal trichromat. are found to be consistent with the simple “self-screening” (bleaching) hypothesis stimulating only three visual pigments, each in its own cone species, so long as it is assumed that the concentrations of these pigments are not dilute. Alpern (1979) rejects the earlier suggestion made by Ingling (1969) that the changes in color matching can be explained by postulating the existence of a yet unidentified second photolabile pigment in the red-receptor mechanism (long-wave cones). He also rejects the idea of Wright’s (1936) that unspecified chemical changes take place in the structure of erythrolabe at higher levels. It is the purpose of this study to test the hypothesis that bleaching of the photopigments can indeed predict the color matches made at a high level of retinal illuminance from those made at a moderate level. The Maxwell method of determining color-matching functions was preferred over the maximum-saturation method because it eliminates effects that may arise when for different test wavelengths simultaneous changes are made in both the test stimulus mixture and the comparison or reference stimulus as is done in the maximum-saturation method. Also it allows the stimulus level to which the color-matching functions apply to be uniquely specified. EXPERIMENTAL
DATA
For a given test stimulus L, presented together with two appropriate primary stimuli in the lower half of the bipartite field, a color match with w (upper half of bipartite field) was fust made at 1000 td (= 1 ktd). The field luminance was then raised so as to provide a retinal illuminance of 2 ktd. This was accomplished by means of a variable sector disk rotating in front of the photometer cube of the trichromator. The radiances of both halves of the bipartite field increased simultaneously and without any wavelength bias. The observer was then given time to adapt to the higher level before adjusting the color match. After a satisfactory match was recorded, the next level was set at 3 ktd and the matching procedure was repeated.
For most test stimuli t involved in the study (with wavelengths corresponding to wavenumbers ranging from 14.500 cm- ’ to 24,500 cm-’ at 250 cm-’ intervals) color matches at the following 17 levels of retinal illuminance (ktd) were made: 1. 2, 3. 5, 7, IO. 14, 18, 24, 30. 37. 45, 55, 65, 75, 85 and 100 ktd. After going from one level to the next, appropriate periods of time (2-5 min) were allowed for adaptation of the observer’s eye to the higher level. The adaptation was considered sufficiently stabilized when repeated color matches did not appear to drift systematically in a certain direction. For each color match, made by the observer at the different levels, the Maxwell tristimulus values were recorded in terms of nanowatts (nW). Within the uncertainties of the observed values, strict proportionality of the tristimulus values was noticed when the level was raised from 1 through 7 ktd and then again when it was raised from 55 through 100 ktd. Between 7 and 55 ktd the tristimulus values increased nonlinearly. However. the magnitude of the deviations from proportionality was highly dependent on the wavelength of the test stimulus L involved in the color match. With L in the long- and middle-wavelength region of the spectrum, quite large breakdowns of the color matches were observed. whereas virtually no breakdowns occurred in the short-wavelength region. These results confirmed those obtained in the pilot study referred to above. In order to test the bleaching hypothesis of the photopigments, we derived two sets of Maxwell spectral distribution functions from the experimental data. The first set represents color matching at moderate levels of retinal illuminance within which the proportionality law holds strictly. This set was derived from the color matches obtained at 1. 2, 3 and 5 ktd. The Maxwell tristimulus values corresponding to the 2, 3 and 5 ktd levels were proportionally scaled down so as to correspond to tristimulus values obtained at 1 ktd. The tristimulus values of the four subsets. all corresponding to observations made at 1 ktd were then averaged. A slight smoothing of the mean Maxwell spectral distribution functions w-as made as well as strict normalization at the primary wavenumbers at which spectral identity between top and bottom halves of the visual field occurred by design. Figure 2 shows the final data representative of Maxwell color matches made at 1 ktd by observer GW. The second set represents color matching at high levels of retinal illuminance within which the proportionality law again holds strictly. This set was derived in a similar way to that used for the first set. However, this time the tristimulus values observed at 65, 75, 85 and 100 ktd were scaled proportionally up to correspond to tristimulus values observed at 100 ktd. The mean values were again slightly smoothed and then normalized at the primary wavenumbers. Figure 3 shows the final data representative of Maxwell color
High-level trichromatic color matching
400
25
450
24
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WAVELENGTH 500
21
20
WAVENUM~~
X hm) 550
I6
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600
I7
650
16
?M3
15
14.103
km-‘)
Fig. 3. Maxwell spectral distribution functions I$.), R(i), G(i) and B(ri)obtained by observer GW in a 2” bipartite visual Reid for two levels of retinal iliuminance: lines refer to data obtained at 1000td (I ktdl, thin lines through open circies refer to data obtained at 100,000 tds (100 ktd).
matches made at 100 ktd by observer GW, compared with those at 1 ktd. This, figure clearly reveals the large differences in ~istimu~us values that are observed with test stimuti in the long and middle region of the spectrum. These differences are indicative of the rather pronounced color differences that are perceived when a color match made at a moderate level breaks down at a high level.
BLEACHING
HYPOTHESIS
We adopt the simple model in which color matching is determined by the absorptions in three classes of receptors containing respectively visual pigments designated “red” (erythrolabe), “green” (chlorolabe) and “blue” (cyanolabe). In Maxwell matching at a given level of the “white” reference stimulus, the total number of quanta absorbed in the visual pigment of each of the receptors of a particular class is determined by the receptor’s visual pigment absorption factor as a function of wavenumber and the absolute spectral radiant power distribution of the stimulus reaching the receptors.
We assume that two different test-stimulus mixtures match the fixed reference stimulus if the total number of quanta absorbed per receptor is the same for both mixtures in receptors (assumed identical) of each of the three classes. Thus two test-stimulus mixtures with external spectral radiant power distributions [J’r(m) dm3 and [P&n) dml given in terms of nanowatts, reference stimulus of
s
wiil color match the
s
pdm) Pt(m) 7(m)QR(m) dm = r(m)rdm) dm II m mm (3)
and two similar equations with a&m) and adm) respectively replacing an(m). Here aR(m), a&m) and aa are respectively the pigment absorption factors for receptors belonging to the three classes, and t(m) is the spectral transmittince of the ocular media accounting for light losses which occur in the eye before the stimuli reach the receptors.
3
G.
WYSZECKI
By the characteristic property of Maxwell colormatching functions the two test-stimulus mixtures also satisfy the equations
i Pl(m)i-(m)
*m
dm = * P&n)tim) dm ! nr
and W. S. STILES
segment. Thus. for [P,(m)dm] we have
the
test-stimulus
mixture
(J) . = const. _/;, [%0(s) - c,,(s)1 ds
(6)
A similar equation holds for the stimulus mixture and since, at color match, the expressions on the left are equal for the two equations, the same is true for the expressions on the right. Hence, [P,(m)dm],
rs
and two simiiar,equations with $m) and 6(m) respectively in place of qtm). In equations (3) it is assumed that the absorption factors ze(m), q-(m), zdm) do not depend upon the test-stimulus mixtures [P,(m)dm] and [Pz(m)dm] that match the reference stimulus. For the simple model in which the outer segment of the receptor is taken to contain only one absorbing substance-the “red”, “green” or “blue” visuat pigment--the following reasoning justifies the assumption made. Suppose, in a retina adapted to darkness, the concentration of the completely unbleached visual pigment in the outer segment varies along the segment according to a function of the distance s from the junction with the inner segment. This function may be designated c&s) in the “red” receptors, where s lies between zero anti S, the length of the outer segment. If, in the steady state, a test mixture [Pl(m)dm] color matches the reference stimulus, the concentration of the unbleached pigment in the “red” outer segments of the test area will be reduced to a different function +i(s). For another test mixture [P2(m)dm] that also color matches the reference stimulus, the corresponding concentration function c&s) will not in genera1 be the same as cR1(s).The absorption factors for the “red” receptors in the two cases will be, respectivefy,
and
where kR is a constant independent of wavenumber, and pCm) is the spectral absorption coefficient of the “red” visual pigment, normalized to unity at the wavenumber m(R) where the coefficient has its greatest value. The total number of quanta of all wavenumbers absorbed per unit time in the outer segment must equal the rate of restoration of complete (unbleached) molecules, which the work of Rushton (f972) has shown to be proportional to the total number of bleached pigment molecules in the receptor, that is to say, the number of protein moieties of originally unbleached pigment, which may be thought of as anchored in the structure of the lamellae of the outer
/ 00
-5
c,,(s)
ds =
c&s) ds = CR.
(7)
0
It follows from the above formulae for JR,(m) and sLR2(m)that these reduce to the same function zR(m) applicable to any stimulus mixture that color matches the reference stimulus. A similar argument defines CG and CB for “green” and “blue” receptors. In actua1 receptors the situation is complicated by the possible presence in appreciable amounts of absorbing substances other than the visual pigment and the effect of these on the visual pigment absorg tion factor for mixture stimuli color matching the same reference stimulus may be different. However. the assumption that the function r,(m) is independent of the relative spectral power distribution of the stimulus mixture will be adopted as a working approximation. This is consistent with the assumed validity of the limited additivity law for Maxwell matching. The two sets of three equations (3) and (4) then imply that the absorption factors m(m). (x&m) and ze(m) are linear combinations of the Maxwell colormatching functions for the fixed reference stimulus when these functions are expressed in quantum units and corrected for prereceptor eye losses. It is convenient for Iater comparisons to define what may be called the “pigment fundamentals” fa(m), f&m) and JB(m) as the spectral absorption factors normalized to unity at their maximum values at wavenumbers m(R), m(G) and m(B), respectively. Thus
=
/l&n) =
fe(m)
r$$c,,i(m) f cizG(m)+ c13b(m)J 1 _
lo-Dc;iml
I-
IO-DG
=
$j
=
’ ,‘“~~~
Cc2 ii(m) +
1
ctd(m)
C c&m)J
= T~Cc31i(m) f c3dm) + c,,&n)] .,...,
(8)
High-level trichromatic color matching tral transmittances
where DR =
WR; DG1: k,C,; D8 =
k,C,
are the total visual pigment densities of the outer segments at their maximai values in the receptors of the three cfasses. The pigment densities DR, DC, Dg,the transformation coefficients Gil, the observed color-matching functions f(m), If(m), 6(m)and the pigment fundamentals fR(m),fG(m) andf&nX ail depend, in general, to some degree on the reference stimulus, and, in particular, vary as the IeveI of the reference stimulus of given relative spectral power distribution is raised from moderate to high values. The absorption coefficient functions, on the other hand. given by the equations
Am) = -d_log[l R
-(1
- 10-DR)~~(m)l
f&4
(9)
1
3
r(m) of the ocular media that is dm)
= -fRff(m) m
‘)
(13
f In the second step, we transform ~~~(rnA~~~~rn~ and f&(m) to predicted dolor-matching functions i*(m)_ J"(m), 6,,(m) normalized at the primaries such that at wavenumbers ma = 15,500 cm-‘, m,s = 19,000 cm-’ and mB = 22,500 cm- * these functions obtain values Of
i
r(m) = -
& log [l
- (1 - lo-D”)~&)3
(10) t
This is readily accomplished by making the transformation matrix (bi&) of
BON= - j+-log [i - (I - 10-Da)~a(m)] J B
represent invariable, intrinsic properties of the three visual pigments. With known absorption coefficients Pfm), y(m) and B(m) we can calculate the pigment fundamentalsfa(m), fc(m) andfe(m) for different pigment densities DR, DG and De. In particular, we can try to predict the pigment fundamentals when bleaching of one or more of the photopigments has occurred as hypothesized in our high-level color-matching experiments. If the pigment densities DRH, DGH and DBH at the high level (symbolized by the subscript N) of retinal illuminance (100 ktd) were known, we could determine j&(m), _#&(m) and jBH(m) from
1 _
fR&)
=
lo-Daw(~l
equal to the inverse of the matrix _fRdmc)
fRfAmd
fGfhG)
fGH(d
j;ssbG)
fidmd
The Maxwell spectral distribution functions E"(m), &(m), G,(m) and B,(m) are then obtained from i,(m), &(m) and 6&m) by means of the reverse transformation of equations (2). Three subsets of transformations are required: (a) Spectral region with which test stimulus .places red primary R (i.e. &(m) = 0):
L re-
1 - 10-D””
&f(m)= &VI - -Mm)Mm)
(14)
&f(m) = &H - L(m) 6&) fsr(m)s
1 - lO-DaHfi(* 1 -
l()‘O’”
By reversing the procedure used above, the pigment fungus f&(m), j&(m) and j&(m) at the high level are readily transformed to color-matching functions and Maxwell spectral distribution functions. In the first step, we revert to pigment fundamentals expressed in energy units and moduiated by the spec-
(b) Spectral region within which test stimulus L replaces green primary C (i.e. G,(m) = 0):
L(m)= -Z$
30
G.
WYSZECKI
(c) Spectral region within which test stimulus places blue primary B (i.e. B,(m) = 0):
Calm)= Gw” - hm)Gdm).
L re-
and LV.S.
STILES
There are different procedures one might follow to obtain an acceptable best set of parameters. One. perhaps most attractive from the mathematical point of Lisw. is to optimize all the parameters simultaneously by. for example. minimizing the following expression:
I
If the hypothesis holds that bleaching of the photopigments accounts for the breakdowns in the color matches observed at 100 ktd retinal illuminance, it must be possible to assign appropriate values to the following parameters that govern the bleaching hypothesis as outlined above:
which may be called the ‘relative standard error”. It contains the sums of squares of the relative differences The tranformation coefficients cilr in equations (8) between predicted and observed hlaxwell distribution which convert the 1 ktd level color-matching funcvalues taken over all wavenumbers m at which obsertions i(m). B(m) and 6(m) to the relative spectral vations were made. In our case. there were n = 35 distributionsf,(m).fc(m) and&(m) of the pigment wavenumbers (not counting the three wavenumbers of fundamentals. the primaries at which the data were normalized). The pigment densities DR, DG and Ds at the I ktd A computer program can be developed to minimize level and DRH. DGH and DBH at the 100 ktd level. S by iterating an initial set of parameters estimated to The spectral transmittance r(m) of the ocular be near the best set. This we tried. but, because of the media. (For details see equation (18). introduced large number of parameters to be optimized in a syslater.) tem of simultaneous nonlinear equations, we found it difficult to ascertain whether a computed minimum S When inserted in equations (2-16). the appropriate values of the above parameters are then expected to was. in fact. the lowest minimum available. Other initransform the experimentally determined Maxwell tial sets of parameters often guided the computer to spectral distribution functions L(m), R(m),G(m) and other minima of S. It appears that in order to obtain B(m). characterizing color matches with a given the desired lowest minimum, while keeping the par“white” reference stimulus at a retinal illuminance of ameters within given constraints, it becomes necessary 1 ktd, to new Maxwell spectral distribution functions to find beforehand an initial set of parameters which virtually has to coincide with the unknown optimal E,,(m). d,(m).G,(m) and B,(m) which agree with the reduces the experimentally determined functions L,(m), R,(m), set. Clearly, this apparent requirement value of the computer program considerably. G,(m) and B,(m) characterizing color matches with We found that a more successful procedure was to the same given “white” reference stimulus but at a use a less ambitious optimization technique. Instead retinal illuminance of 100 ktd. of having the computer iterate all parameters in a Because of the nature of the experiment and its given sequence of operation, we iterated the parinherent uncertainties in both the observations and ameters individually, observing at all times the effect a instrumental calibrations, we cannot expect to obtain small change in one parameter had on the value of S. perfect agreement between predicted and observed both in magnitude and direction. We also recorded Maxwell spectral distribution functions. However, we the effect such a change had on the relative differences should expect to find one or more sets of parameters between the predicted and observed Maxwell distrieach of which would provide an agreement within the bution values as a function of wavenumber. This range of the experimental uncertainties. Such a set enabled us to apply different weights to the residual difmight be called an “optimal” or “best” set of parferences in different regions of the spectrum. Finally. ameters. we monitored the relative spectral distributions of the Another important requirement is to constrain the pigment fundamentals as they were affected by iteraparameters within certain “acceptable” ranges. The tions of the c,~ coefficients. This helped us in contransformation coefficients cik should provide relative straining the cilr coefficients so as to keep the positions spectral distributions of the pigment fundamentals of the peaks of the pigment fundamentals as well as which resemble those expected from other studies, the shape of these curves to within reasonable such as the Kiinig-type fundamentals or the n-mechanisms. The optical densities DR, Dc and D, of the bounds. The whole of the procedure is admittedly not a very unbleached photopigments should fall within a elegant one from the mathematical point of view, and reasonable range of values, somewhere between 0.2 certainly required considerable human judgement in and 1.0. The spectral transmittances r(m)of the ocular addition to computer time. However, the procedure media should agree reasonably well with those found converged to a set of parameters which, while meeting in normal eyes.
31
High-level trichromatic color matching
leads to S = 0.016. However, the improvement gained was considered too small and thus the case was not pursued further. The agreement between predicted and observed 100 ktd Maxwell spectral distribution functions is quite good when we made allowances for the inherent uncertainties in the observation data both at the 1 ktd and the 100 ktd levels of retinal illuminance. In Figs 4-7 we have plotted the ratios of the predicted over the observed MaxwelI spectra1 distribution functions. that is,
certain conditions of constraint. provided a satisfactorily low relative standard error S. RESC’LTS
Our best set of parameters obtained by the procedure described above is the following: (a) Coeficienrs ci,: I.353.t0-s 3.223. lo-’ 2.098. lO-6 3.954. IOqs 7.771’ 10-E l.813-10-6 (b) Piymenr densities: DR = 0.44 D, = 0.38 DRH= 0.01 D,, = 0.01 (c) Spectral
transmittance
3.008~ lo+’ 3.035. IO+ I. 148. lO-5 DB =
0.45
L(m) &(m) t%(m) J,(m) L,(m,‘R,(m)‘GH(m)‘BH(n)
DBH = 0.44 r(m) of ocultrr media: The
inert pigments in the eye’s lens and macula are initially assumed to have spectral transmittances t,_(m) and r,,,(m), respectively, in accordance with data usually applied to the normal eye (Wy~ecki and Stiles, 1967). However, as the densities of these pigments may differ from individual to individual, we introduced parameters 6 and E designed to allow for such density difference between the normal eye and observer GW. Thus we considered spectral transmittances s(m) of the ocular media to be composed of spectral transmittances r;(m) of the lens pigment and spectral transmittances &(m) of the macular pigment; that is r(m) = T~(rn)‘~~~(rn)
Perfect agreement between the predicted and observed data would result in ratios of 1.00 at all wavenumbers. This, of course, does not occur, except at the wavenumbers of the primary stimuli R, G and B. However, at these wavenumbers the agreement has been forced to be perfect through the required normalization of the data as described above. The discrepancies are, with few exceptions, considered small and most likely can be attributed to the random and systematic uncertainties inherent in the observations and instrumental calibrations. A comprehensive error analysis is rather difficult. However, we performed a crude analysis which provides some quantitative insight into the error problem. From repeat observations, instrumental calibrations, and by taking account of the fact that some minor smoothing of the raw observational data was made, we estimate that our Maxwell color matches [f.(m), R(m), G(m) and B(m)] at 1 ktd agree with the “true” color matches within +27& while those at 100 ktd [L.,(m), R,(m), G,(m) and B”(m)] agree within +4%. These estimates are probably conservative rather than excessive and, of course, apply only to observer GW and the NRC trichromator used in this study. Errors in the observed 100 ktd Maxwell color matches affect the ratios plotted in Figs 4-7 in a
(18)
Our best case required b = 1.00 and E = 0.80 that is, normal density of the lens pigment, but only 80% of the normal density of the macular pigment. With the parameters given under (a), (b) and (c) above, the relative standard error S (equation 17) becomes S = 0.018 or 1.80,; A small reduction of S can still be obtained by further reducing the density of the macular pigment to E = 0.75 of its normal density, while keeping all other parameters unchanged. This
1.06 1.04 ?
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Fig. 4. Ratio &&r)&,(m) of predicted over observed 100 ktd Maxwell values as function of wavenumber m. Hatched area represents uncertainty range of ratios when assuming t2% errors in the observed I ktd Maxwell matches and +4/, errors in the observed 100 ktd Maxwell matches.
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I 20
I I8 t 16 I 14
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WAVENUMBER
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Fig. 7. Ratio &(m)/&(m) of predicted over observed 100 ktd Maxwell values as function of wavenumber m. Hatched area represents uncertainty range of ratios when assuming +20, errors in the observed 1 ktd Maxwell matches and 14% errors in the observed 100 ktd Maxwell matches.
High-level trichromatic color matching
33
log scale. The shapes of these curves are in fair agreestraightforward way. Increasing (or decreasing) an ment with those obtained by observer GW in 1978 by observed value - bv* 4”/& decreases (or increases) the the maximum-~turation method, using the same ~orres~nding ratio by 44;. For exampte. if, at a given wavenumber m. L,(m) is changed to L;I = 1.04 instrument and keeping the retinal iiluminance at ap proximately 1 ktd for all test stimuli. The latter da& L,,(m). the ratio ~,(m)/L&) changes to &(m)/L&) which is 4% lower than the original ratio. It follows are shown by the broken lines in Figs 8a-c. Some of that the upper and lower bounds, beyond which the the differences between the two sets of data may be attributable to instrumental and observational uncerdiscrepancies shown in Figs 4-7 become significant, tainties. On the other hand. some differences may be are simply horizontal lines at 1.04 and 0.96, respectattributable to the use of the maximum-saturation ively. method rather than the Maxwell method. Errors in the observed t ktd Maxweli color However, we note one qualitative peculiarity in the matches propagate through the calculations made to arrive at the predicted 100 ktd Maxwell values. It is long-wave spectral region of the 6(m)function. Instead ,not immediately obvious how large an effect of a of finding an ail-negative lobe between the c and R +2% error has on t”(m), B”(m). C,(m) and t?,(m),primaries and a positive lobe beyond the R primary, but a systematic numerical exploration provided us we obtain a lobe that is partly positive and partly negative between G and f? and all-negative beyond K. with the desired upper and lower bounds of the effect. The procedure was first to change by +2x, for a This, at first, was puzzling; but when we considered given wavenumber m, the 1 ktd values of f.(m), R(m), the effect +2% errors in the corresponding observed G(m)and B(m)one at a time and then recalculate the values of L(m),R(m),G(m) and B(m) in this region predicted 100 ktd values E,(m),i?"(m). G,(m) and would have on the 6(m) function, it became clear that the 6(m) function in this region cannot be determined B,(m). These calculations resulted in new predicted 100 ktd Maxwell values which differed from the origi- with the certainty required to be sure of its shape. The shaded areas in Figs 8a-c around the mean nal values. Positive as well as negative differences curves of the color-matching functions indicate the occurred and their magnitude varied with m. When uncertainty range which we must expect from f29;; we changed the 1 ktd values by -2% virtually identical differences were obtained except their signs had errors in the observed I ktd Maxwell spectral distribution functions. In general, the major lobes of the reversed. From these differences we calculated upper F(m). g(m)and 6(m)functions are obtained with fair and lower bounds of the 100 ktd Maxwefl values. certainty whereas the minor lobes are associated with The upper and lower bounds are best illustrated when transferred to the ratio graphs in Figs 4-7. Here quite large uncertainty ranges, especially near the prithey have been added to the 4% bounds assumed for mary positions. Figure 8d illustrates, on an expanded linear scale, the effect on the 6(m) function beyond the the observed 100 ktd Maxwell values. The upper and G primary. lower bounds shown in Figs 4-7 thus define the range From equations (8) we derive the relative spectral within which the ratios of predicted and observed distributions fR(m), f&m),_&(m)of the fnn~mentals. 100 ktd Maxwell values can fall if the observed 1 ktd and observed 100 ktd MaxweII values are assumed to These transformations involve the l ktd color-matching functions and the coefficients Cikas defined in our be uncertain by +2% and &4% respectively. We note best set of parameters. Figures 9a-c illustrate fR(m) this uncertainty range is not independent of waver(m)andfB(m) r(m), that is the relative specnumber. Near the wavenumbers m = 18,OOOcm-’ r(m),&(m) tral distributions of the pigment fundamentals uncorand 20,250 cm- ’ the uncertainty range becomes very rected for light losses in the ocular media, but large. Near these wavenumbers the 100 ktd color expressed in quantum units. The shaded areas again match with the “white” reference stimulus requires represent the uncertainty ranges expected from i2”/, only two (rather than the normal three) non-zero errors postulated in the observed 1 ktd Maxwell specstimuli (see Fig. 3). tral distribution functions. Fairly large uncertainty Except for a few isolated wavenumbers, the ratios ranges occur in the short-wave spectral regions of the of predicted over observed 100 ktd Maxwell values “red” and “green” pigment fundamentals and in the generally fall well inside the estimated uncertainty range. This indicates that our “best set” of parlong-wave spectral region of the “blue” pigment fundamental. ameters, which governs the bleaching hypothesis, predicts 100 ktd Maxwell matches within the uncertainty The pigment fundamentals yielded by the bleaching of our experimental data. However, what remains to hypothesis are specifically defined in terms of the be considered is whether our parameters also meet model envisaging a single visuai pigment in each the requirements of being “acceptable- from other receptor. Other sets of “fundamentals” can be derived points of view, as discussed above. from color-matching data on the basis of Kiinig’s hyThe 1 ktd color-matching functions qrnh am) and pothesis that the three types of dichromar (prota6(m) are derived from the observed 1 ktd Maxwell nopes. deuteranopes and tritanopes) possess reduced spectral distribution functions L(m),R(m),G(m) and forms of trichromatic vision but each lacking one-a B(m) by means of equations (2). Figures 8a-c show the different one-of three normal visual response sysderived 1 ktd color-matching functions plotted on a terns. The spectral sensitivities of the distinct response
31
G.
WYYECKS
systems. the fundamentals, are derivable from the color-matching functions of the normal trichromat together with the locations in the uichromatic chromaticity diagram of the confusion points of the three types of dichromat. Good examples of such sets of fundamentals are those derived by Vos and Walraven (I971), Wafraven (1974) and Smith et al. (1976). A further possibility is that the spectral sensitivities
and W. S.
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35
High-level trichromatic color matching
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of the z-mechanisms (Stiles, 1978) might be identified as pigment fundamentals. These sensitivities were determined by methods not involving color matching and aim to represent the spectral sensitivities of distinct retinal response systems each of which might “A 20/1--c
involve more than one visual pigment. The work of Pugh (1976) makes it virtually certain that as regards the field sensitivity of the x1 or blue-sensitive mechanism at longer wavelengths (2 > 5 10 nmb “green” or “red” or both visual pigments are involved as well as
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Fig. 10a.b.c. Relative spectral distributions of "red". “green” and “blue” fundamentals: (i) Curve through solid circles: pigment fundamentalsJ,(m).f,(m) andfs(m) as predicted in this study. (ii) Curve through open circles: field sensitivities of Stiles n-mechanisms (n,. n, and n,). (iii) Curve through crosses: Kijnig-type fundamentals proposed by Smith et al. (1976). (iv) Dotted curve: Konig-type fundamentals proposed by Vos and Walraven (Walraven. 174). Ali functions are normalized to unity at their respective peaks. are given in quantum units. and are corrected for light losses in the ocular media.
High-level trichromatic color matching the “blue” pigment. Nevertheless. the field sensitivities
of JQ, nL and A~ have been found to be expressible approximately as linear combinations of the colormatching functions (Estevez and Cavonius. 1977; Pugh and Sigel, 1978). Figures 1Oa-c compare our fundamentals derived in the course of testing the bleaching hypothesis, with those based on the n-mechanisms and those of the Kiinig type suggested by Vos and Walraven (see Walraven. 1974, for latest version) and by Smith et al. (1976). Only relative spectral distributions of the fundamentals are shown, that is. all have been normalized to have unit values at the wavenumbers at which their respective peaks occur. All functions are given in terms of quantum units and are corrected for light losses in the ocular media. In our case, r(m) is defined by equation (18) with 6 = 1.00 and E = 0.80. In the case of the n-mechanisms and the Vos-Walraven fundamentals we have set 6 = Q = 1.0, and in the case of the Smith et al. fundamentals we have used 6 = 1.33 and E = 0.50 in accordance with the values set by Smith er al. In a qualitative sense, there is reasonable agreement between the various fundamentals derived for the “red”-receptor mechanism and an equally reasonable agreement between those derived for the “green”receptor mechanism. Our prediction of the “red” and “geen” fundamentals fall comfortably in the domain of the corresponding fundamentals derived by the other investigators. A great deal of the somewhat larger spread between corresponding fundamentals in the short-wave region of the spectrum can most certainly be attributed to differences in the densities of the ocular media assumed to apply to the different data. However, to assess the significance of these differences and those of a more subtle nature in quantitative terms is difficult as we tack knowledge of the true fundamentals. A somewhat less reasonable agreement is noted between our “blue” fundamental and those of the other investigators. The case of the blue fundamentals is intriguing because our prediction demands a pronounced secondary peak in the long-wave region of the spectrum. There is a great deal of flexibility in the shape of this secondary peak, which is governed by the coefficients cII and cJ2 in the transformation of equations (8). The upper (continuous) curve corresponds to our best case. The lower (broken) curve corresponds to coefficients c~, = 6.084. IO-* and cj2 = 3.012.10-‘. These new coefficients, with all others kept unchanged. have only a marginal effect on the relative standard error S. defined by equation (17): S increases from 0.018 to 0.020. However, they have a drastic effect on the long-wave part of the “blue” fundamental. There is an indication that the n, mechanisms also has a secondary peak in that spectral. region. but it is much less pronounced and is only barely caught on our graph. The good agreement between the 1c, mechanism and the Smith et al. “blue” fundamental is not surprising because Smith et ccl.chose the
37
n, mechanism as their model for the “blue” fundamental. With regard to the densities of the photopigments we believe that our estimates for the “red” and “green” pigments fall within expected ranges. Our estimate for the “blue” pigment. however, is virtually uncertain because that pigment did not seem to bleach by any measurable amount in our experiment. It follows that the given densities of DB = 0.45 and DBH = 0.44 could well be given any other values, provided that the near unity ratio of DBH/DB is approximately kept the same. For example, the use of DB = 0.100 and DBH = 0.098 in our best set of parameters gives essentially identical results. The reason for the somewhat unexpected behavior of the “blue” pigment is not clear. At this time, we can only speculate. We suggest that the “blue” pigment may be in a near-diluted state already at moderate levels of retinal illuminance, making it difficult to detect its bleaching at higher levels. Acknowledyemenrs-The authors wish to thank Mr G. H. Fielder for operating, calibrating and maintaining the trichromator during these lengthy experiments. and for carrying out numirous calcu~at~ons.~ Thanks are also extended to Dr D. S. Richie for writing some of the complex computer programs required for the optimization techniques explored during this study. REFERENCES
Alpern M. (1979)Lack of uniformity in colour matching. J. Ph_kol. 288. 85- 105. Brindley G. S. (1953) The effects on color vision of adaptation to very bright lights. J. Physiol. 122, 332-350. Crawford B. H. (1965) Colour matching and adaptation. Vision Rex 5, 71-78.
Estevez 0. and Cavonius C. R. (1977) Human color perception and Stiles’ a mechanisms. Vision Res. 17, 417-422.
Ingling C. R. Jr (1969) A tetrachromatic
hypothesis for human color vision. Vision Res. 9, 1131-I I%l. Lozano R. D. and Palmer D. A. (1968) Large-field color matching and adaptation. J. opr. Sot. Am. 58. 1653-1656. Pugh E. N. Jr (1976) The nature of the coiour mechanisms of W. S. Stiles. J. Phvsiol. 257. 713-747. Pugh E. N. Jr and Sigel C. (1978) Evaluation of the candidacy of the n-mechanisms of Stiles for color-matching fundamentals. Vision Res. 18, 317-330. Rushton W. A. H. (1972) Visual Pigments in Man. In: Handbook of&nsory Physiology, Vol. VII,l, pp. 36&394. Springer, New York. Smith V. C., Pokorny J. and Starr S. (1976) Variability of color mixture data-I. Interobserver variability in the unit coordinates. Vision Res. 16, 1087-1094. Stiles W. S. (1978) Mechanisms of Co/our vision. Academic Press. London. Terstiege H. (1967) Untersuchungen zum Persistenz-und Koeffizientensatz. Die Farbe 16. I-I 20. Vos J. J. and Walraven P. L. (1971) On the derivation of the fovea1 receptor primaries. Vision Res. 11, 799-818. Walraven P. L. (1974) A closer look at the tritanouic convergence point. Vision Res. 14. i339-1343. . Wright W. D. (1936) The breakdown of a colour match w:th high in&sit& of adaptation. J. Physiol. 87, 23-33. Wyszecki G. (1978) Color matching at moderate to high levels of retinal illuminance: A pilot study. Vision Res. 18. 34 l-346. Wyszecki G. and Stiles W. S. (1967) Color Science. Wiley, New York.
W. S. STILES 89 Richmond Hill Court. Richmond. Surrey. England (Receiced 20 April 1979) color matches with reference to a given “white” stimulus at a retinal illuminante of 100,000 td can be predicted from Maxwell-type color matches made with reference to the same “white” stimulus but at a retinal illuminance of loo0 td by means of a simple hypothesis of bleaching of the visual pigments. The spectral distributions of the fundamentals of color vision. the densities of the visual pigments of the 1000 and 100,000 td levels, as well as the spectral transmittances of the lens and macula pigments which must be assumed as essential parameters of the visual mechanism emerge with values which appear consistent with those obtained from other investigations. While both the “red” and the “green” fundamentals exhibit strong and predictable bleaching characteristics, the “blue” fundamental appears to bleach only very little under the given experimental conditions. The somewhat unexpected behavior of the “blue” fundamental is not explained, but it is suggested that the pigment associated with the “blue” fundamental may be in a near-diluted state already at moderate levels of retinal illuminance, making it difficult to detect bleaching at higher levels. Abstract-Maxwell-type
the B primary so that the mixture [L(/.)L + B(i)B] color matches the mixture [R(i.)R- + G(i)G]. When the amount of the desaturating B primary is very small. it is sometimes helpful to add a small amount of the B primary to the upper half of the bipartite field and then take as the match the difference in the amounts of the B primaries in the two halves. The amounts R(i). G(i). B(L) of t,he three primaries required in the match are called the tristimulus values of the test stimulus L(i.)L. The amount of the desaturating primary is given a negative sign. Apart from differences at test stimuli of short wavelengths consistent with different absorptions in the eye lens and macular pigment, the respective color matches of normal trichromatic observers agree rather closely, leading to similar tristimulus values R(i), G(i) and B(i) for a given stimulus L(i)L. From such data the familiar color-matching functions t+.), a(;.), 6(i) emerge:
INTRODUCTION Color matching is one of the most fundamental psychophysical experiments in color vision. Color-matching data provide the basis for color vision theories as well as the basis for calorimetry. There are two main methods that have been used in color-matching experiments: the maximum-satu:ation method and the Maxwell method. Figure 1 gives the parameters which are essential to the two methods. A bipartite visual field is usually used in either method. In the illustrated case the field has a horizontal dividing line. There are three fixed primary stimuli R, C, B which are normally monochromatic stimuli consisting of narrow bands of wavelength i, one in the red, one in the green and one in the blue part of the spectrum. Then there is a variable test stimulus L, normally also monochromatic but of variable wavelength i., and finally a fixed reference stimulus w, normally of a “white” color. The maximum-saturation method shown on the left of Fig. 1 is the most commonly used method. The test stimulus L, of magnitude L(L). presented in the lower half of the bipartite field, is “desaturated” by the minimum amount required of one of the three primary stimuli R, G, B such that the mixture of the test stimulus and the desaturating primary can be color matched by an appropriate mixture of the remaining two primaries appearing in the upper half of the bipartite field. The choice of the desaturating primary depends solely on the wavelength of the test stimulus relative to the wavelengths of the primaries. In the case illustrated, the test stimulus L is desaturated by
F(i) = R(i.)/L(i.) g(i) = G(i.)jL(i.) 6(i) = B(i.)/L(i)
(1)
The Maxwell method shown on the right of Fig. 1 employs a fixed reference stimulus m, normally of a “white” color, appearing in the upper half of the bipartite field. In the lower half, a mixture of the test stimulus L and two of the available three primary stimuli (R, G and B) is presented. which when appropriate amounts of them a& chosen, color matches the reference stimulus. The wavelength of the test stimulus determines which two of the three primaries are 23
G. %‘YSzfCKl snd ‘A’. s.
Fig. i. Color matching in a bipartite visual field using the maximum-saturation me&hod or the lMaxwell method. R, G. B = Primary stimuli (of fixed wavelengths i,. i.,. i.,) R(b). G(i). S(L) = Amounts of primary stimufi (tristimuhts values) (e.g. in terms of radiance: W m - z sr - ’ ) g = Variable test stimulus of wavelength E. L(L) = Amount of variable test stimulus (e.g. in terms of radiance: W rneL St--‘) @ = Fixed reference stimulus (e.g. of “white” color) W = Amount of fixed reference stimulus (e.g. in luminance or trolands)
added to the test stimulus. In the example illustrated, the stimulus WV color matches the mixture [R(EJR + G(L)c + t(i.)L]. As in the maximum-saturation method, normal trichromatic observers agree
rather closeiy with their respective color matches, leading to similar tristimulus values. The results of a complete color-matching experiment by means of the Maxwell method can be illustrated graphically as shown in Fig. 2. From this graph one can read off the tristimulus values t(i), R(i.), G(i) and B(E.)of each test stimulus L of wavelength i. or wavenumber nr in terms of the radiant power manowatts, nW) of the monochromatic stimuli involved. For example. when the test stimulus L is set at m = 16,500 cm- ’ (or i 2 606 nm), a mixture of L, G and B using amounts L(L) = 0.399, p;iL) = 0.355 and Bfi.) = 0.344 nW, respectivefy. are required by the observer to obtain a color match with the reference stimulus w at 1000 td. At wavenumbers m = 17,6OOcm-’ and m s 20,3OOcm-’ we notice certain singularities in the Maxwell spectral distribution curves of Fig. 2. When the test stimutus L is set at these wavenumbers only one, instead of two, of the three primary stimuli is required to provide. when mixed with .E, a color match with @. With L at 17,6GQcm- ’ the addition of B suffices, and with L at 20,300 cm- * the addition of I? suffices. These results are quite expected and in accordance with the laws of trichromatic color matching.
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Fig 2. Maxwell spectral distribution functions L.(i.).R(i.). G(i.) and B(L) obtained by observer CW in a 2” bipartite visual field. The reference “white” stimulus is of constant radiance providing a retinal illuming ance of loo0 td (1 ktd) and consists of a mixture of the three instrumental primaries R. 6 and B of radiant powers equal to 1.0856. 0.5567 and 0.3515nW. respectively.
25
High-level trichromatic color matching
The data shown in Fig. 2 were obtained by observer GW (then of age 51) on NRC’s trichrornator. The observing conditions and instrumental parameters were similar to those described in the pilot study published earlier (Wyszecki, 1978): Visuuffiefd:
2’ bipartite circular field with dark sur-
round, seen in Maxwellian view. The entrance slits of the three monochromators which make up the trichromator are imaged in the center of the natural pupil. These images are small enough (not more than 2 mm dia in any direction) to be unobstructed by the eye’s iris diaphragm. Instrumental primaries (R, G, B): f? centered at wavenumber ma = 15,5OOcm-’ (z 645 nm) G centered at wavenumber mG = 19,000 cm- l ( 5 526nm) II centered at wavenumber ma = 22,5OOcm-’ (z 444nm) The width Am of the waveband of each primary stimulus was the same and set at 500 cm- I. Test stimuli (L): L centered at wavenumbers
rangfrom m = 14,500 cm- ’ (369Onm) to m = 24,5OOcm-’ (t 408 nm) at regular intervals of 250 cm-i. The width Am of the waveband varied with the wavenumber m of the test stimulus. However, in using three different slits when traversing the long-, middle- and short-wavelength regions of the spectrum, it was possible to have wavebands Am = 5OOcm-’ when m was equal to mR, mc or ma. Thus, whenever the test stimulus L was centered at mR = 15,5oOcm-‘, mG = 19,OOOcm- or ma = 22,500 cm- ‘, an appropriate radiance adjustment was all that was needed to obtain a complete physical match between L and one of the primary stimuli, R. G or B. Reference stimulus (w): Stimulus of average daylight color with (x, y)-chromaticity of CIE standard illuminant D6s. The reference stimulus was produced by an appropriate mixture of the instrumental primaries l?, G and 8. To obtain a retinal illuminance of 1000 td, the required radiant powers for the three primaries were ing
Rw = 1.0856nW
Gw = 0.5567 nW Bw = 0.3516 nW The fact that w is an additive mixture of the three instrumental primaries anchors the Maxwell spectral distribution curves of Fig. 2 at the wavenumbers of the primaries l?, G and B as indicated by solid circles. That is, whenever the test stimulus L has a wavenumber identical’to that of a primary stimulus, both the reference stimulus w, presented in the upper half of the bipartite visual field, and the additive mixture of stimuli presented in the lower haif of the field become physically identical as soon as a color match is produced by the observer.
The results of the color-matching experiment by means of the Maxwell method as illustrated in Fig. 2 can be converted to the familiar color-matching functions i(;.), d(;.b 6(i.). The following equations are used: i(j) = g(i.) =
&.) =
Rw - R(i) L(j.) Gw - G(L)
(2)
L(i) Bw - i?(i.) L(i)
I
As can be seen from Fig. 2, L(i) never becomes zero, whereas R(i.), G(1) and B(1) are respectively zero in the long-, middle- and short-wavelength region of the spectrum. The quantities Rw, Gw and Bw are fixed throughout the experiment and define the color of the reference stimulus w. It is clear that if the basic laws of trichromatic color matching, particularly the proportionality and additivity laws, hold strictly, the color-matching functions ?(i.), g(1) and 6(i) derived from Maxwell matches in accordance with equations (2) must (within the experimental uncertainties) be identical to those derived more directly by means of the maximum-saturation method (equations 1). However, there is experimental evidence (for example, Crawford, 1965; Lozano and Palmer, 1968) of systematic differences between the two sets of color-matching functions obtained by the two different methods. In this study we will assume that color matching done at a given level of retinal illuminance and with a fixed relative spectral composition of the “white” reference stimulus ‘strictly obeys the color-matching laws. With these restrictions the laws reduce to asserting a more limited additivity law, namely, that if any two test stimulus mixtures A and B match individually the fixed reference stimulus, then so does any additive mixture of A and B in amounts pA and (1 - p) B respectively where p is less than unity. Observations by Lozano and Palmer (1968) using a 10’ matching field suggest that even this limited additivity law may not be strictly obeyed. But a critical study for a 2” field has not yet been made and in what follows the validity of the limited additivity law will be assumed. Our attention will be directed to a failure in the proportionality law of color matching when the level of retinal illuminance is changed from -a moderate level (1000 td) to a high level (100,000 td), the relative spectral composition of the reference stimulus being kept the same. As described in the pilot study (Wyszecki, 1978), some color matches made at 1000 td clearly do not hold when viewed at 100,000 td. In fact, the breakdown of the proportionality law was observed to set in at approx 8000 td and appeared to be completed at approx 50,000 td. Above 50,000 td the proportionality
26
G. WYSZECKI and W. S. STILES
law appeared to hold again. For some color matches. the magnitude of the breakdown can be quite large when the 50,OtMltd level is reached. It is generally assumed that the breakdowns of the color matches caused by increases in the retinal illuminance can be attributed to bleaching of the photopigments, particularly the pigment (or pigments) assumed to reside in the “red-receptor mechanism” (Wright, 1936; Brindley. 1953; Terstiege, 1967: Ingling. 1969). In a thorough study just published by Alpern (1979), it is shown that the fraction of the red stimulus in a red-green mixture of stimuli. colormatched to a yellow stimulus. increases when the level of stimulation, in a Maxwell-type experiment. is sufficiently increased to bleach the “red” (erythrolabe) and the “green” (chlorolabe) pigments. The changes in color matching in the yellow region of the spectrum, observed by a normal trichromat. are found to be consistent with the simple “self-screening” (bleaching) hypothesis stimulating only three visual pigments, each in its own cone species, so long as it is assumed that the concentrations of these pigments are not dilute. Alpern (1979) rejects the earlier suggestion made by Ingling (1969) that the changes in color matching can be explained by postulating the existence of a yet unidentified second photolabile pigment in the red-receptor mechanism (long-wave cones). He also rejects the idea of Wright’s (1936) that unspecified chemical changes take place in the structure of erythrolabe at higher levels. It is the purpose of this study to test the hypothesis that bleaching of the photopigments can indeed predict the color matches made at a high level of retinal illuminance from those made at a moderate level. The Maxwell method of determining color-matching functions was preferred over the maximum-saturation method because it eliminates effects that may arise when for different test wavelengths simultaneous changes are made in both the test stimulus mixture and the comparison or reference stimulus as is done in the maximum-saturation method. Also it allows the stimulus level to which the color-matching functions apply to be uniquely specified. EXPERIMENTAL
DATA
For a given test stimulus L, presented together with two appropriate primary stimuli in the lower half of the bipartite field, a color match with w (upper half of bipartite field) was fust made at 1000 td (= 1 ktd). The field luminance was then raised so as to provide a retinal illuminance of 2 ktd. This was accomplished by means of a variable sector disk rotating in front of the photometer cube of the trichromator. The radiances of both halves of the bipartite field increased simultaneously and without any wavelength bias. The observer was then given time to adapt to the higher level before adjusting the color match. After a satisfactory match was recorded, the next level was set at 3 ktd and the matching procedure was repeated.
For most test stimuli t involved in the study (with wavelengths corresponding to wavenumbers ranging from 14.500 cm- ’ to 24,500 cm-’ at 250 cm-’ intervals) color matches at the following 17 levels of retinal illuminance (ktd) were made: 1. 2, 3. 5, 7, IO. 14, 18, 24, 30. 37. 45, 55, 65, 75, 85 and 100 ktd. After going from one level to the next, appropriate periods of time (2-5 min) were allowed for adaptation of the observer’s eye to the higher level. The adaptation was considered sufficiently stabilized when repeated color matches did not appear to drift systematically in a certain direction. For each color match, made by the observer at the different levels, the Maxwell tristimulus values were recorded in terms of nanowatts (nW). Within the uncertainties of the observed values, strict proportionality of the tristimulus values was noticed when the level was raised from 1 through 7 ktd and then again when it was raised from 55 through 100 ktd. Between 7 and 55 ktd the tristimulus values increased nonlinearly. However. the magnitude of the deviations from proportionality was highly dependent on the wavelength of the test stimulus L involved in the color match. With L in the long- and middle-wavelength region of the spectrum, quite large breakdowns of the color matches were observed. whereas virtually no breakdowns occurred in the short-wavelength region. These results confirmed those obtained in the pilot study referred to above. In order to test the bleaching hypothesis of the photopigments, we derived two sets of Maxwell spectral distribution functions from the experimental data. The first set represents color matching at moderate levels of retinal illuminance within which the proportionality law holds strictly. This set was derived from the color matches obtained at 1. 2, 3 and 5 ktd. The Maxwell tristimulus values corresponding to the 2, 3 and 5 ktd levels were proportionally scaled down so as to correspond to tristimulus values obtained at 1 ktd. The tristimulus values of the four subsets. all corresponding to observations made at 1 ktd were then averaged. A slight smoothing of the mean Maxwell spectral distribution functions w-as made as well as strict normalization at the primary wavenumbers at which spectral identity between top and bottom halves of the visual field occurred by design. Figure 2 shows the final data representative of Maxwell color matches made at 1 ktd by observer GW. The second set represents color matching at high levels of retinal illuminance within which the proportionality law again holds strictly. This set was derived in a similar way to that used for the first set. However, this time the tristimulus values observed at 65, 75, 85 and 100 ktd were scaled proportionally up to correspond to tristimulus values observed at 100 ktd. The mean values were again slightly smoothed and then normalized at the primary wavenumbers. Figure 3 shows the final data representative of Maxwell color
High-level trichromatic color matching
400
25
450
24
23
22
WAVELENGTH 500
21
20
WAVENUM~~
X hm) 550
I6
19
600
I7
650
16
?M3
15
14.103
km-‘)
Fig. 3. Maxwell spectral distribution functions I$.), R(i), G(i) and B(ri)obtained by observer GW in a 2” bipartite visual Reid for two levels of retinal iliuminance: lines refer to data obtained at 1000td (I ktdl, thin lines through open circies refer to data obtained at 100,000 tds (100 ktd).
matches made at 100 ktd by observer GW, compared with those at 1 ktd. This, figure clearly reveals the large differences in ~istimu~us values that are observed with test stimuti in the long and middle region of the spectrum. These differences are indicative of the rather pronounced color differences that are perceived when a color match made at a moderate level breaks down at a high level.
BLEACHING
HYPOTHESIS
We adopt the simple model in which color matching is determined by the absorptions in three classes of receptors containing respectively visual pigments designated “red” (erythrolabe), “green” (chlorolabe) and “blue” (cyanolabe). In Maxwell matching at a given level of the “white” reference stimulus, the total number of quanta absorbed in the visual pigment of each of the receptors of a particular class is determined by the receptor’s visual pigment absorption factor as a function of wavenumber and the absolute spectral radiant power distribution of the stimulus reaching the receptors.
We assume that two different test-stimulus mixtures match the fixed reference stimulus if the total number of quanta absorbed per receptor is the same for both mixtures in receptors (assumed identical) of each of the three classes. Thus two test-stimulus mixtures with external spectral radiant power distributions [J’r(m) dm3 and [P&n) dml given in terms of nanowatts, reference stimulus of
s
wiil color match the
s
pdm) Pt(m) 7(m)QR(m) dm = r(m)rdm) dm II m mm (3)
and two similar equations with a&m) and adm) respectively replacing an(m). Here aR(m), a&m) and aa are respectively the pigment absorption factors for receptors belonging to the three classes, and t(m) is the spectral transmittince of the ocular media accounting for light losses which occur in the eye before the stimuli reach the receptors.
3
G.
WYSZECKI
By the characteristic property of Maxwell colormatching functions the two test-stimulus mixtures also satisfy the equations
i Pl(m)i-(m)
*m
dm = * P&n)tim) dm ! nr
and W. S. STILES
segment. Thus. for [P,(m)dm] we have
the
test-stimulus
mixture
(J) . = const. _/;, [%0(s) - c,,(s)1 ds
(6)
A similar equation holds for the stimulus mixture and since, at color match, the expressions on the left are equal for the two equations, the same is true for the expressions on the right. Hence, [P,(m)dm],
rs
and two simiiar,equations with $m) and 6(m) respectively in place of qtm). In equations (3) it is assumed that the absorption factors ze(m), q-(m), zdm) do not depend upon the test-stimulus mixtures [P,(m)dm] and [Pz(m)dm] that match the reference stimulus. For the simple model in which the outer segment of the receptor is taken to contain only one absorbing substance-the “red”, “green” or “blue” visuat pigment--the following reasoning justifies the assumption made. Suppose, in a retina adapted to darkness, the concentration of the completely unbleached visual pigment in the outer segment varies along the segment according to a function of the distance s from the junction with the inner segment. This function may be designated c&s) in the “red” receptors, where s lies between zero anti S, the length of the outer segment. If, in the steady state, a test mixture [Pl(m)dm] color matches the reference stimulus, the concentration of the unbleached pigment in the “red” outer segments of the test area will be reduced to a different function +i(s). For another test mixture [P2(m)dm] that also color matches the reference stimulus, the corresponding concentration function c&s) will not in genera1 be the same as cR1(s).The absorption factors for the “red” receptors in the two cases will be, respectivefy,
and
where kR is a constant independent of wavenumber, and pCm) is the spectral absorption coefficient of the “red” visual pigment, normalized to unity at the wavenumber m(R) where the coefficient has its greatest value. The total number of quanta of all wavenumbers absorbed per unit time in the outer segment must equal the rate of restoration of complete (unbleached) molecules, which the work of Rushton (f972) has shown to be proportional to the total number of bleached pigment molecules in the receptor, that is to say, the number of protein moieties of originally unbleached pigment, which may be thought of as anchored in the structure of the lamellae of the outer
/ 00
-5
c,,(s)
ds =
c&s) ds = CR.
(7)
0
It follows from the above formulae for JR,(m) and sLR2(m)that these reduce to the same function zR(m) applicable to any stimulus mixture that color matches the reference stimulus. A similar argument defines CG and CB for “green” and “blue” receptors. In actua1 receptors the situation is complicated by the possible presence in appreciable amounts of absorbing substances other than the visual pigment and the effect of these on the visual pigment absorg tion factor for mixture stimuli color matching the same reference stimulus may be different. However. the assumption that the function r,(m) is independent of the relative spectral power distribution of the stimulus mixture will be adopted as a working approximation. This is consistent with the assumed validity of the limited additivity law for Maxwell matching. The two sets of three equations (3) and (4) then imply that the absorption factors m(m). (x&m) and ze(m) are linear combinations of the Maxwell colormatching functions for the fixed reference stimulus when these functions are expressed in quantum units and corrected for prereceptor eye losses. It is convenient for Iater comparisons to define what may be called the “pigment fundamentals” fa(m), f&m) and JB(m) as the spectral absorption factors normalized to unity at their maximum values at wavenumbers m(R), m(G) and m(B), respectively. Thus
=
/l&n) =
fe(m)
r$$c,,i(m) f cizG(m)+ c13b(m)J 1 _
lo-Dc;iml
I-
IO-DG
=
$j
=
’ ,‘“~~~
Cc2 ii(m) +
1
ctd(m)
C c&m)J
= T~Cc31i(m) f c3dm) + c,,&n)] .,...,
(8)
High-level trichromatic color matching tral transmittances
where DR =
WR; DG1: k,C,; D8 =
k,C,
are the total visual pigment densities of the outer segments at their maximai values in the receptors of the three cfasses. The pigment densities DR, DC, Dg,the transformation coefficients Gil, the observed color-matching functions f(m), If(m), 6(m)and the pigment fundamentals fR(m),fG(m) andf&nX ail depend, in general, to some degree on the reference stimulus, and, in particular, vary as the IeveI of the reference stimulus of given relative spectral power distribution is raised from moderate to high values. The absorption coefficient functions, on the other hand. given by the equations
Am) = -d_log[l R
-(1
- 10-DR)~~(m)l
f&4
(9)
1
3
r(m) of the ocular media that is dm)
= -fRff(m) m
‘)
(13
f In the second step, we transform ~~~(rnA~~~~rn~ and f&(m) to predicted dolor-matching functions i*(m)_ J"(m), 6,,(m) normalized at the primaries such that at wavenumbers ma = 15,500 cm-‘, m,s = 19,000 cm-’ and mB = 22,500 cm- * these functions obtain values Of
i
r(m) = -
& log [l
- (1 - lo-D”)~&)3
(10) t
This is readily accomplished by making the transformation matrix (bi&) of
BON= - j+-log [i - (I - 10-Da)~a(m)] J B
represent invariable, intrinsic properties of the three visual pigments. With known absorption coefficients Pfm), y(m) and B(m) we can calculate the pigment fundamentalsfa(m), fc(m) andfe(m) for different pigment densities DR, DG and De. In particular, we can try to predict the pigment fundamentals when bleaching of one or more of the photopigments has occurred as hypothesized in our high-level color-matching experiments. If the pigment densities DRH, DGH and DBH at the high level (symbolized by the subscript N) of retinal illuminance (100 ktd) were known, we could determine j&(m), _#&(m) and jBH(m) from
1 _
fR&)
=
lo-Daw(~l
equal to the inverse of the matrix _fRdmc)
fRfAmd
fGfhG)
fGH(d
j;ssbG)
fidmd
The Maxwell spectral distribution functions E"(m), &(m), G,(m) and B,(m) are then obtained from i,(m), &(m) and 6&m) by means of the reverse transformation of equations (2). Three subsets of transformations are required: (a) Spectral region with which test stimulus .places red primary R (i.e. &(m) = 0):
L re-
1 - 10-D””
&f(m)= &VI - -Mm)Mm)
(14)
&f(m) = &H - L(m) 6&) fsr(m)s
1 - lO-DaHfi(* 1 -
l()‘O’”
By reversing the procedure used above, the pigment fungus f&(m), j&(m) and j&(m) at the high level are readily transformed to color-matching functions and Maxwell spectral distribution functions. In the first step, we revert to pigment fundamentals expressed in energy units and moduiated by the spec-
(b) Spectral region within which test stimulus L replaces green primary C (i.e. G,(m) = 0):
L(m)= -Z$
30
G.
WYSZECKI
(c) Spectral region within which test stimulus places blue primary B (i.e. B,(m) = 0):
Calm)= Gw” - hm)Gdm).
L re-
and LV.S.
STILES
There are different procedures one might follow to obtain an acceptable best set of parameters. One. perhaps most attractive from the mathematical point of Lisw. is to optimize all the parameters simultaneously by. for example. minimizing the following expression:
I
If the hypothesis holds that bleaching of the photopigments accounts for the breakdowns in the color matches observed at 100 ktd retinal illuminance, it must be possible to assign appropriate values to the following parameters that govern the bleaching hypothesis as outlined above:
which may be called the ‘relative standard error”. It contains the sums of squares of the relative differences The tranformation coefficients cilr in equations (8) between predicted and observed hlaxwell distribution which convert the 1 ktd level color-matching funcvalues taken over all wavenumbers m at which obsertions i(m). B(m) and 6(m) to the relative spectral vations were made. In our case. there were n = 35 distributionsf,(m).fc(m) and&(m) of the pigment wavenumbers (not counting the three wavenumbers of fundamentals. the primaries at which the data were normalized). The pigment densities DR, DG and Ds at the I ktd A computer program can be developed to minimize level and DRH. DGH and DBH at the 100 ktd level. S by iterating an initial set of parameters estimated to The spectral transmittance r(m) of the ocular be near the best set. This we tried. but, because of the media. (For details see equation (18). introduced large number of parameters to be optimized in a syslater.) tem of simultaneous nonlinear equations, we found it difficult to ascertain whether a computed minimum S When inserted in equations (2-16). the appropriate values of the above parameters are then expected to was. in fact. the lowest minimum available. Other initransform the experimentally determined Maxwell tial sets of parameters often guided the computer to spectral distribution functions L(m), R(m),G(m) and other minima of S. It appears that in order to obtain B(m). characterizing color matches with a given the desired lowest minimum, while keeping the par“white” reference stimulus at a retinal illuminance of ameters within given constraints, it becomes necessary 1 ktd, to new Maxwell spectral distribution functions to find beforehand an initial set of parameters which virtually has to coincide with the unknown optimal E,,(m). d,(m).G,(m) and B,(m) which agree with the reduces the experimentally determined functions L,(m), R,(m), set. Clearly, this apparent requirement value of the computer program considerably. G,(m) and B,(m) characterizing color matches with We found that a more successful procedure was to the same given “white” reference stimulus but at a use a less ambitious optimization technique. Instead retinal illuminance of 100 ktd. of having the computer iterate all parameters in a Because of the nature of the experiment and its given sequence of operation, we iterated the parinherent uncertainties in both the observations and ameters individually, observing at all times the effect a instrumental calibrations, we cannot expect to obtain small change in one parameter had on the value of S. perfect agreement between predicted and observed both in magnitude and direction. We also recorded Maxwell spectral distribution functions. However, we the effect such a change had on the relative differences should expect to find one or more sets of parameters between the predicted and observed Maxwell distrieach of which would provide an agreement within the bution values as a function of wavenumber. This range of the experimental uncertainties. Such a set enabled us to apply different weights to the residual difmight be called an “optimal” or “best” set of parferences in different regions of the spectrum. Finally. ameters. we monitored the relative spectral distributions of the Another important requirement is to constrain the pigment fundamentals as they were affected by iteraparameters within certain “acceptable” ranges. The tions of the c,~ coefficients. This helped us in contransformation coefficients cik should provide relative straining the cilr coefficients so as to keep the positions spectral distributions of the pigment fundamentals of the peaks of the pigment fundamentals as well as which resemble those expected from other studies, the shape of these curves to within reasonable such as the Kiinig-type fundamentals or the n-mechanisms. The optical densities DR, Dc and D, of the bounds. The whole of the procedure is admittedly not a very unbleached photopigments should fall within a elegant one from the mathematical point of view, and reasonable range of values, somewhere between 0.2 certainly required considerable human judgement in and 1.0. The spectral transmittances r(m)of the ocular addition to computer time. However, the procedure media should agree reasonably well with those found converged to a set of parameters which, while meeting in normal eyes.
31
High-level trichromatic color matching
leads to S = 0.016. However, the improvement gained was considered too small and thus the case was not pursued further. The agreement between predicted and observed 100 ktd Maxwell spectral distribution functions is quite good when we made allowances for the inherent uncertainties in the observation data both at the 1 ktd and the 100 ktd levels of retinal illuminance. In Figs 4-7 we have plotted the ratios of the predicted over the observed MaxwelI spectra1 distribution functions. that is,
certain conditions of constraint. provided a satisfactorily low relative standard error S. RESC’LTS
Our best set of parameters obtained by the procedure described above is the following: (a) Coeficienrs ci,: I.353.t0-s 3.223. lo-’ 2.098. lO-6 3.954. IOqs 7.771’ 10-E l.813-10-6 (b) Piymenr densities: DR = 0.44 D, = 0.38 DRH= 0.01 D,, = 0.01 (c) Spectral
transmittance
3.008~ lo+’ 3.035. IO+ I. 148. lO-5 DB =
0.45
L(m) &(m) t%(m) J,(m) L,(m,‘R,(m)‘GH(m)‘BH(n)
DBH = 0.44 r(m) of ocultrr media: The
inert pigments in the eye’s lens and macula are initially assumed to have spectral transmittances t,_(m) and r,,,(m), respectively, in accordance with data usually applied to the normal eye (Wy~ecki and Stiles, 1967). However, as the densities of these pigments may differ from individual to individual, we introduced parameters 6 and E designed to allow for such density difference between the normal eye and observer GW. Thus we considered spectral transmittances s(m) of the ocular media to be composed of spectral transmittances r;(m) of the lens pigment and spectral transmittances &(m) of the macular pigment; that is r(m) = T~(rn)‘~~~(rn)
Perfect agreement between the predicted and observed data would result in ratios of 1.00 at all wavenumbers. This, of course, does not occur, except at the wavenumbers of the primary stimuli R, G and B. However, at these wavenumbers the agreement has been forced to be perfect through the required normalization of the data as described above. The discrepancies are, with few exceptions, considered small and most likely can be attributed to the random and systematic uncertainties inherent in the observations and instrumental calibrations. A comprehensive error analysis is rather difficult. However, we performed a crude analysis which provides some quantitative insight into the error problem. From repeat observations, instrumental calibrations, and by taking account of the fact that some minor smoothing of the raw observational data was made, we estimate that our Maxwell color matches [f.(m), R(m), G(m) and B(m)] at 1 ktd agree with the “true” color matches within +27& while those at 100 ktd [L.,(m), R,(m), G,(m) and B”(m)] agree within +4%. These estimates are probably conservative rather than excessive and, of course, apply only to observer GW and the NRC trichromator used in this study. Errors in the observed 100 ktd Maxwell color matches affect the ratios plotted in Figs 4-7 in a
(18)
Our best case required b = 1.00 and E = 0.80 that is, normal density of the lens pigment, but only 80% of the normal density of the macular pigment. With the parameters given under (a), (b) and (c) above, the relative standard error S (equation 17) becomes S = 0.018 or 1.80,; A small reduction of S can still be obtained by further reducing the density of the macular pigment to E = 0.75 of its normal density, while keeping all other parameters unchanged. This
1.06 1.04 ?
1.02
3
1.00
\
O.%
2
0.46
if
0.94 0.92 0.90 0.98 L
I
,
25
24
23
0
, .22
21
20
WAVENUM9ER
E
I
L
,
19
18
17
16
R
, IS
14. Id
m (cm-‘)
Fig. 4. Ratio &&r)&,(m) of predicted over observed 100 ktd Maxwell values as function of wavenumber m. Hatched area represents uncertainty range of ratios when assuming t2% errors in the observed I ktd Maxwell matches and +4/, errors in the observed 100 ktd Maxwell matches.
G.
WYSZECKi
and
!H.
s. STILES
________A_____
0.92 0.90 L
0 88
t
a
2%i-&-
22
21
20
WAVENUMBER
‘5
I9
I6
ii 1;
16
I;
t 14 IO’
m (cm“)
Fig. 5. Ratio &,(m)/&,(m) of predicted over observed 100 ktd Maxwell values as function of wavenumber m. Hatched area represents uncertainty range of ratios when assuming &I!“, errors in the observed I ktd Maxwell matches and i-4:~; errors in the observed 100 ktd Maxweil matches.
I .06
I.w zx I.02 c I.00 T. 098 fZ
0.96 094 0.92
;
0.90
I
1 I 25
1
24
23
22
L
!
G -
,
21
20
19
I8
WAVENUMBER
_.
17
I6
I5
14. IO’
m km-‘)
Fig. 6. Ratio ~~(rn)~G~(rn) of predicted over observed 100 ktd %LIaxwellvalues as function of wavenumber m. Hatched area represents uncertainty range of ratios when assuming rtt?; errors in the observed I ktd Maxwell matches and k44; errors in the observed 100 ktd Maxwell matches.
I 20
I I8 t 16 I 14
25
24
23
22
21
20
WAVENUMBER
.tY
16
I7
I6
t5
m km”)
Fig. 7. Ratio &(m)/&(m) of predicted over observed 100 ktd Maxwell values as function of wavenumber m. Hatched area represents uncertainty range of ratios when assuming +20, errors in the observed 1 ktd Maxwell matches and 14% errors in the observed 100 ktd Maxwell matches.
High-level trichromatic color matching
33
log scale. The shapes of these curves are in fair agreestraightforward way. Increasing (or decreasing) an ment with those obtained by observer GW in 1978 by observed value - bv* 4”/& decreases (or increases) the the maximum-~turation method, using the same ~orres~nding ratio by 44;. For exampte. if, at a given wavenumber m. L,(m) is changed to L;I = 1.04 instrument and keeping the retinal iiluminance at ap proximately 1 ktd for all test stimuli. The latter da& L,,(m). the ratio ~,(m)/L&) changes to &(m)/L&) which is 4% lower than the original ratio. It follows are shown by the broken lines in Figs 8a-c. Some of that the upper and lower bounds, beyond which the the differences between the two sets of data may be attributable to instrumental and observational uncerdiscrepancies shown in Figs 4-7 become significant, tainties. On the other hand. some differences may be are simply horizontal lines at 1.04 and 0.96, respectattributable to the use of the maximum-saturation ively. method rather than the Maxwell method. Errors in the observed t ktd Maxweli color However, we note one qualitative peculiarity in the matches propagate through the calculations made to arrive at the predicted 100 ktd Maxwell values. It is long-wave spectral region of the 6(m)function. Instead ,not immediately obvious how large an effect of a of finding an ail-negative lobe between the c and R +2% error has on t”(m), B”(m). C,(m) and t?,(m),primaries and a positive lobe beyond the R primary, but a systematic numerical exploration provided us we obtain a lobe that is partly positive and partly negative between G and f? and all-negative beyond K. with the desired upper and lower bounds of the effect. The procedure was first to change by +2x, for a This, at first, was puzzling; but when we considered given wavenumber m, the 1 ktd values of f.(m), R(m), the effect +2% errors in the corresponding observed G(m)and B(m)one at a time and then recalculate the values of L(m),R(m),G(m) and B(m) in this region predicted 100 ktd values E,(m),i?"(m). G,(m) and would have on the 6(m) function, it became clear that the 6(m) function in this region cannot be determined B,(m). These calculations resulted in new predicted 100 ktd Maxwell values which differed from the origi- with the certainty required to be sure of its shape. The shaded areas in Figs 8a-c around the mean nal values. Positive as well as negative differences curves of the color-matching functions indicate the occurred and their magnitude varied with m. When uncertainty range which we must expect from f29;; we changed the 1 ktd values by -2% virtually identical differences were obtained except their signs had errors in the observed I ktd Maxwell spectral distribution functions. In general, the major lobes of the reversed. From these differences we calculated upper F(m). g(m)and 6(m)functions are obtained with fair and lower bounds of the 100 ktd Maxwefl values. certainty whereas the minor lobes are associated with The upper and lower bounds are best illustrated when transferred to the ratio graphs in Figs 4-7. Here quite large uncertainty ranges, especially near the prithey have been added to the 4% bounds assumed for mary positions. Figure 8d illustrates, on an expanded linear scale, the effect on the 6(m) function beyond the the observed 100 ktd Maxwell values. The upper and G primary. lower bounds shown in Figs 4-7 thus define the range From equations (8) we derive the relative spectral within which the ratios of predicted and observed distributions fR(m), f&m),_&(m)of the fnn~mentals. 100 ktd Maxwell values can fall if the observed 1 ktd and observed 100 ktd MaxweII values are assumed to These transformations involve the l ktd color-matching functions and the coefficients Cikas defined in our be uncertain by +2% and &4% respectively. We note best set of parameters. Figures 9a-c illustrate fR(m) this uncertainty range is not independent of waver(m)andfB(m) r(m), that is the relative specnumber. Near the wavenumbers m = 18,OOOcm-’ r(m),&(m) tral distributions of the pigment fundamentals uncorand 20,250 cm- ’ the uncertainty range becomes very rected for light losses in the ocular media, but large. Near these wavenumbers the 100 ktd color expressed in quantum units. The shaded areas again match with the “white” reference stimulus requires represent the uncertainty ranges expected from i2”/, only two (rather than the normal three) non-zero errors postulated in the observed 1 ktd Maxwell specstimuli (see Fig. 3). tral distribution functions. Fairly large uncertainty Except for a few isolated wavenumbers, the ratios ranges occur in the short-wave spectral regions of the of predicted over observed 100 ktd Maxwell values “red” and “green” pigment fundamentals and in the generally fall well inside the estimated uncertainty range. This indicates that our “best set” of parlong-wave spectral region of the “blue” pigment fundamental. ameters, which governs the bleaching hypothesis, predicts 100 ktd Maxwell matches within the uncertainty The pigment fundamentals yielded by the bleaching of our experimental data. However, what remains to hypothesis are specifically defined in terms of the be considered is whether our parameters also meet model envisaging a single visuai pigment in each the requirements of being “acceptable- from other receptor. Other sets of “fundamentals” can be derived points of view, as discussed above. from color-matching data on the basis of Kiinig’s hyThe 1 ktd color-matching functions qrnh am) and pothesis that the three types of dichromar (prota6(m) are derived from the observed 1 ktd Maxwell nopes. deuteranopes and tritanopes) possess reduced spectral distribution functions L(m),R(m),G(m) and forms of trichromatic vision but each lacking one-a B(m) by means of equations (2). Figures 8a-c show the different one-of three normal visual response sysderived 1 ktd color-matching functions plotted on a terns. The spectral sensitivities of the distinct response
31
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35
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of the z-mechanisms (Stiles, 1978) might be identified as pigment fundamentals. These sensitivities were determined by methods not involving color matching and aim to represent the spectral sensitivities of distinct retinal response systems each of which might “A 20/1--c
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High-level trichromatic color matching the “blue” pigment. Nevertheless. the field sensitivities
of JQ, nL and A~ have been found to be expressible approximately as linear combinations of the colormatching functions (Estevez and Cavonius. 1977; Pugh and Sigel, 1978). Figures 1Oa-c compare our fundamentals derived in the course of testing the bleaching hypothesis, with those based on the n-mechanisms and those of the Kiinig type suggested by Vos and Walraven (see Walraven. 1974, for latest version) and by Smith et al. (1976). Only relative spectral distributions of the fundamentals are shown, that is. all have been normalized to have unit values at the wavenumbers at which their respective peaks occur. All functions are given in terms of quantum units and are corrected for light losses in the ocular media. In our case, r(m) is defined by equation (18) with 6 = 1.00 and E = 0.80. In the case of the n-mechanisms and the Vos-Walraven fundamentals we have set 6 = Q = 1.0, and in the case of the Smith et al. fundamentals we have used 6 = 1.33 and E = 0.50 in accordance with the values set by Smith er al. In a qualitative sense, there is reasonable agreement between the various fundamentals derived for the “red”-receptor mechanism and an equally reasonable agreement between those derived for the “green”receptor mechanism. Our prediction of the “red” and “geen” fundamentals fall comfortably in the domain of the corresponding fundamentals derived by the other investigators. A great deal of the somewhat larger spread between corresponding fundamentals in the short-wave region of the spectrum can most certainly be attributed to differences in the densities of the ocular media assumed to apply to the different data. However, to assess the significance of these differences and those of a more subtle nature in quantitative terms is difficult as we tack knowledge of the true fundamentals. A somewhat less reasonable agreement is noted between our “blue” fundamental and those of the other investigators. The case of the blue fundamentals is intriguing because our prediction demands a pronounced secondary peak in the long-wave region of the spectrum. There is a great deal of flexibility in the shape of this secondary peak, which is governed by the coefficients cII and cJ2 in the transformation of equations (8). The upper (continuous) curve corresponds to our best case. The lower (broken) curve corresponds to coefficients c~, = 6.084. IO-* and cj2 = 3.012.10-‘. These new coefficients, with all others kept unchanged. have only a marginal effect on the relative standard error S. defined by equation (17): S increases from 0.018 to 0.020. However, they have a drastic effect on the long-wave part of the “blue” fundamental. There is an indication that the n, mechanisms also has a secondary peak in that spectral. region. but it is much less pronounced and is only barely caught on our graph. The good agreement between the 1c, mechanism and the Smith et al. “blue” fundamental is not surprising because Smith et ccl.chose the
37
n, mechanism as their model for the “blue” fundamental. With regard to the densities of the photopigments we believe that our estimates for the “red” and “green” pigments fall within expected ranges. Our estimate for the “blue” pigment. however, is virtually uncertain because that pigment did not seem to bleach by any measurable amount in our experiment. It follows that the given densities of DB = 0.45 and DBH = 0.44 could well be given any other values, provided that the near unity ratio of DBH/DB is approximately kept the same. For example, the use of DB = 0.100 and DBH = 0.098 in our best set of parameters gives essentially identical results. The reason for the somewhat unexpected behavior of the “blue” pigment is not clear. At this time, we can only speculate. We suggest that the “blue” pigment may be in a near-diluted state already at moderate levels of retinal illuminance, making it difficult to detect its bleaching at higher levels. Acknowledyemenrs-The authors wish to thank Mr G. H. Fielder for operating, calibrating and maintaining the trichromator during these lengthy experiments. and for carrying out numirous calcu~at~ons.~ Thanks are also extended to Dr D. S. Richie for writing some of the complex computer programs required for the optimization techniques explored during this study. REFERENCES
Alpern M. (1979)Lack of uniformity in colour matching. J. Ph_kol. 288. 85- 105. Brindley G. S. (1953) The effects on color vision of adaptation to very bright lights. J. Physiol. 122, 332-350. Crawford B. H. (1965) Colour matching and adaptation. Vision Rex 5, 71-78.
Estevez 0. and Cavonius C. R. (1977) Human color perception and Stiles’ a mechanisms. Vision Res. 17, 417-422.
Ingling C. R. Jr (1969) A tetrachromatic
hypothesis for human color vision. Vision Res. 9, 1131-I I%l. Lozano R. D. and Palmer D. A. (1968) Large-field color matching and adaptation. J. opr. Sot. Am. 58. 1653-1656. Pugh E. N. Jr (1976) The nature of the coiour mechanisms of W. S. Stiles. J. Phvsiol. 257. 713-747. Pugh E. N. Jr and Sigel C. (1978) Evaluation of the candidacy of the n-mechanisms of Stiles for color-matching fundamentals. Vision Res. 18, 317-330. Rushton W. A. H. (1972) Visual Pigments in Man. In: Handbook of&nsory Physiology, Vol. VII,l, pp. 36&394. Springer, New York. Smith V. C., Pokorny J. and Starr S. (1976) Variability of color mixture data-I. Interobserver variability in the unit coordinates. Vision Res. 16, 1087-1094. Stiles W. S. (1978) Mechanisms of Co/our vision. Academic Press. London. Terstiege H. (1967) Untersuchungen zum Persistenz-und Koeffizientensatz. Die Farbe 16. I-I 20. Vos J. J. and Walraven P. L. (1971) On the derivation of the fovea1 receptor primaries. Vision Res. 11, 799-818. Walraven P. L. (1974) A closer look at the tritanouic convergence point. Vision Res. 14. i339-1343. . Wright W. D. (1936) The breakdown of a colour match w:th high in&sit& of adaptation. J. Physiol. 87, 23-33. Wyszecki G. (1978) Color matching at moderate to high levels of retinal illuminance: A pilot study. Vision Res. 18. 34 l-346. Wyszecki G. and Stiles W. S. (1967) Color Science. Wiley, New York.