BULLETII~" OF MATHEMATICAL BIOPHYSICS VOLUME 26, 1964

A N A N A L Y S I S O F T H E 2 ~ A N D 10 ~ F I E L D

C O L O R - M A T C H I N G F U N C T I O N S

W~TMAN RICHARDS

Psycho logy Section, Massachuse t t s I n s t i t u t e o f Technology,

Cambridge, Mass.

The 2 ~ and 10 ~ field color-matching functions are independent: one specification is not a linear t ransformation of the other, even after correcting for maeular pigment effects. Therefore, the " t rue" color-matching functions which directly describe the linear responses of the eye must be different for the two field sizes. This means tha t a given stimulus will, in general, have a different chromatici ty depending upon the field size, regardless of the choice of any one colorimetrie co-ordinate system for all field sizes. However, in spite of these chromatici ty differences, a large uniform field usually appears near ly uniform. Such color uniformity implies tha t even though chromatic differences occur as a function of ret inal position or field size, these differences are small. I f this is the case, then the underlying " t rue" color-matching functions determining the observed color-matching functions must be nearly, bu t not quite, identical. These differences vanish as ident i ty between the sets of color-matching functions is approached. This p roper ty suggests a method of calculating the " t rue" color-matching fimctions. The " t rue" color-matching functions must approximate those obtained b y minimizing the chromatici ty differences between two independent sets of data. This can be done by assuming tha t the coefficients of t ransformation should be adjusted so as to produce as nearly identical chromaticit ies for spectrum stimuli as possible. I n this paper, i t is also assumed tha t the " t rue" color- matching functions have no negative values, as if they were based on actual absorption spectra. This article describes the calculation of the " t rue" 2 ~ and 10 ~ field color-match- ing functions satisfying these two conditions. For both field sizes, the maxima of the three functions are near 435, 540, and 585 m/z, after correcting for the filtering effects of the ocular media and macular pigment.

T h e e x p e r i m e n t a l l y d e t e r m i n e d 2 ~ a n d 10 ~ f ie ld c o l o r - m a t c h i n g f u n c t i o n s (St i les

a n d B u r c h , 1959) a r e l i n e a r l y i n d e p e n d e n t (St i les , 1958). I f c o r r e c t i o n s a r e

m a d e fo r t h e d i f f e r en t f i l t e r ing effects o f t h e m a c u l a r p i g m e n t for t h e t w o f ie ld

sizes, b o t h spec i f i ca t i ons for co lor m a t c h i n g s t i l l r e m a i n l i n e a r l y i n d e p e n d e n t

9

10 WHITMAN RICHARDS

(Stiles, 1958). There is no linear operator that can transform color-matching functions for the 2 ~ field into color-matching functions for the 10 ~ field. This means that a 2 ~ central region of a 10 ~ uniform chromatic stimulus may appear different from the peripheral part of the 10 ~ field.

The dependence of color-matching data on the visual field size requires that actual wave-length dependent functions, describing the linear responses of the visual process, will change with field size. These actual wave-length dependent functions will be defined as "action spectra." The action spectra represent only the linear responses occurring at some stage in the visual process; they do not necessarily represent actual absorption spectra of photosensitive pigments. Because the 2 ~ and 10 ~ field color-matching functions are different, the action spectra describing the responses for these fields are also different. These differences cannot be the result of ocular filters (such as the maeular pigment), as previously mentioned, or of neural processes in the eye, bu t must be due to differences in the receptor responses.

Associated with the action spectra for any visual field is a triplet of "true" color-matching functions. These true color-matching functions represent the responses of their underlying action spectra after the stimulus has been modi- fied by the effects of ocular filters. Therefore, the " true" color-matching func- tions for the 2 ~ and 10 ~ fields differ in two respects: in their underlying action spectra, and in the effects of their ocular filters. However, in each case, the " true" color-matehing functions are explicitly defined as a linear transforma- tion of the appropriate standard 2 ~ or 10 ~ field color-matching data. The ob- jective of this analysis is to determine the " true" color-matching functions for the 2 ~ and 10 ~ fields, and to calculate representative action spectra, which directly describe the linear responses of one aspect of the visual process.

In order to determine these true response functions, two assumptions must be made. The first is that for a given field size some linear transformation of the empirical color-matching functions describes positive responses to radiant energy, most likely directly related to the absorption of energy by photo- sensitive molecules. In this case, all values of the transformed color-matching functions must be positive. The second assumption is tha t independent chromaticity functions are isomorphic with independent components of the underlying visual process, and that differences in the underlying visual proc- esses result in differences in observed chromaticities. To a first approxima- tion, the relationship is assumed linear for sufficiently small changes. In particular, this assumption states that for a monochromatic stimulus, the changes in the chromaticity functions will reflect changes in the underlying visual process for any sufficiently small change in field size. The simplified case restricted to monochromatic stimuli simplifies the mathematical description of

COLOR-MATCHING FUNCTIONS ]1

an analyzable relationship between the color-matching functions and the underlying visual processes. The generalization to any arbitrary stimulus then follows from Grassman's laws. The unique correlates of wave length in terms of the color-matching functions are two ratios of pairs of the three color-match- ing functions. For any given field size, these pairs of ratio values have a one- to-one correspondence with wave length for any linear transformation of the standard specification. This isomorphic relationship suggests a method for describing color variations. A measure of the invariance of a monochromatic stimulus is the increment in wave-length of the stimulus applied to one field, which is necessary to give identical ratio values for the corresponding pairs of the true color-matching functions for the second field size. But if such incre- ments are always small, the wave-length differences can be approximated by ratio differences. Therefore, the true color-matching functions for each field can be determined by simultaneously minimizing for all wave lengths the differ- ences between the ratio values of the corresponding color-matching functions for both field sizes. Such differences of ratios actually correspond more nearly to just noticeable color differences than do wave-length differences themselves. The use of ratios also has the additional advantage of discounting for the effects of all ocular filters.

Let f~l(~, ~, 5), f~2(~, ?~, 5), and f23(~, ~, ~) represent the three independent "true" color-matching functions for the 2 ~ field in terms of the standard 2 ~ field specification, x, y, z. Because there is a one-to-one correspondence between chromaticity and wave length, there is also a one-to-one correspondence be- tween chromaticity for a 2 ~ field stimulus and the ratios f~l/f~2 and f2a/f92 for any monochromatic stimulus. Similarly, if fl01(e, ~, 5), f102(~, ~7, 5), and flo3(~, ~7, 5) represent the three independent true color-matching functions for the 10 ~ field in terms of the standard 10 ~ field specification, ~, g, 5, then flol/flo2 and flo3/flo2 have a one-to-one correspondence with chromaticity for mono- chromatic stimuli for 10 ~ fields. I f the differences between the 2 ~ and 10 ~ field color-matching functions are not too great, an approximate solution for the true color-matching functions can be obtained by assuming that each ratio value correlates with one of two independent components of the chromatic specification in the retina. This assumption is equivalent to approximating by a linear function an unknown relationship between two possible chromatic components, and should not introduce significant errors in these calculations. In this case, the selection of the true color-matching functions must be such that f2~/f2~ approximate flol/flo9 or that f23/f2~, approximate flo3/flo2 as closely as possible. However, because the constants of transformation determining the f functions are initially unknown, no distinction can be made between any two ratios of functions taken from the same set of data until after an initial solution

12 WHITMAN RICHARDS

has been obtained. Therefore, only one pair of functions taken from each independent triplet of functions needs to be compared simultaneously over the visible spectrum. The simplest condition for comparing two ratios of func- tions, for instance f21/f22 with flo1/flo2, is to compute the quantity:

[f22 f-~oJ / (1)

which sums over all wave lengths the squared differences in the wave-length dependent ratios ful/f22 and flol/flo2. E is the experimental error of the color- matching data and weights the differences calculated at each wave length in terms of the reliability of the data at that wave length. The f functions which minimize condition (1) also minimize the chromaticity differences at mono- chromatic wave lengths between the 2 ~ and 10 ~ field color specifications, and, as discussed above, correspond to the " true" response functions.

The values of E used in the calculations are given in Table I. These values are weighted per cent errors taken from the 10 ~ field mean color-matching data of Stiles and Burch, 1959. At each wave length, the E values are obtained by adding each per cent error times the amount of its primary divided by the sum of all three primary values. These E values are weighting factors of first-order approximation only, and are used primarily to reduce the significance of the comparisons between f21/f22 and flol/flo2 for wave lengths near the ends of the visible spectrum. The procedure for determining the functions f21, f22, flol, f~o2 is to minimize the value of condition (1) by adjusting the coefficients of the s ~, 5, or ~, ~, ~, terms of each function. Therefore, changes in the coefficients which increase the value of condition (1) are discarded. Such calculations were made at 36 equally spaced wave lengths on a Bendix G-15 computer. After successive iterations, the coefficients of the following equations were calculated to the second decimal place, which was the maximum accuracy consistent with the accuracy of the data:

f21 = a2 = 1.50s + 1.097~ + .075 (2)

f22 = 72 = - . 81s + 3 . 0 0 ~ + .225 (3)

f101 = alo = .64~ + 1.01~ + .92~ (4)

f102 = 71o = .11~ + 2.28~ + .27~. (5)

These functions are " true" color-matching functions and represent the responses of the visual process much more closely than the original eolorimetric data. The values between 400-750 m~ at 10 m/~ intervals are given in Table II. The

COLOR.MATCHING FUNCTIONS

T A B L E I

13

W a v e l e n g t h m ~

400 410 420 430 440

450 460 470 480 490

500 510 520 530 540

550 560 570 580 590

600 610 620 630 640

650 660 670 680 690

700 710 720 730 740 750

W e i g h t e d % E r r o r (E)

56 32 16 10

9.0

9.3 11 12 12 11.5

9.5 7.5 5.8 5,3 6.0

7.5 8.5 9.0 9.1 8.6

7.5 6.0 5.0 4.5 4.2

4.2 4.6 5.0 5.7 6.5

7.8 9.0

10.6 12 15 18

% Difference of Resu l t s*

11 16 12

3.4 - - 2.3

- - 7 . 0

- - 14 - -22

24 20

16 8.6 5.1 1.9

- - 1.6

- 4 . 6

- - 7 . 1

- - 9.7 11 10

7.9 5.1 3.3

.7 - - 4 . I

- 1 . 0

- 3.8 - - . 9

1.0 .4

T r a n s m i t t a n c e ( Judd , 1951)

Ocular Media (t)

.572

.583

.594

.602

.610

.619

.631 .641 .649

Macu la r Pigment (/~)

- - 2.8 6.8

- - 7.5 - - 33

29 - -40

.657

.664

.676

.690

.698

.705

.707

.708

.710

.711

.713

.086 .577

.106 .506 .160 .396 .248 .316 .318 .305

.388 .212

.426 .206

.438 .299

.458 .250

.481 .263

.495 .516

.510 .798

.525 .935

.543 .968

.559 .977

.566 .985 .989 .989 .989 .989

1.000 1.000 1.000 1.000 1.000

1.000 1.000 1.000 1.000 1.000

1.000 1.000 1.000 1.000 1 . 0 0 0

1.000

~1o ~ (72 max o

14 WHITMAN RICHARDS

T A B L E I I

Calculated values of the " t r u e " 2 ~ and 10 ~ field color .matching funct ions

Wave length m~

400 410 420 430 440

450 460 470 480 490

500 510 520 530 540

550 560 570 580 590

600 610 620 630 640

650 660 670 680 690

700 710 720 730 740 750

0"2

.0266

.0811

.2512

.5355

.6698

.6698 ,6184 .4824 ,3518 ,3073

.3785 ,5733 ,8743

1.1908 1.4769

1.7353 1.9766 2.1810 2.3229 2.3647

2.2811 2.0522 1.6969 1.2525 .8626

.5419

.3138

.1660

.0887

.0430

.0216

.0110

.0054

.0026

.0014

.0006

~2

.0046

.0140

.0452

.1097

.1712

.2315

.3117

.3980

.5184

.7004

1.0249 1.5363 2.0959 2.4612 2.6312

2.6359 2.5043 2.2392 1.8682 1.4399

1.0328 .6970 .4510 .2747 .1622

.0914

.0494

.0252

.0131

.0062

.0031

.0016

.0007

.0004

.0003

.0001

10 ~ Field Values

.0399

.1857

.4523

.6995

.8704

.8814 ,7784 ,6035 .4364 .3777

.4376

.6102

.8464 1.1072 1.4011

1.6557 1.9107 2.1564 2.2931 2.3750

2.2697 2.0120 1.6316 1.2317 .8424

.5391

.3289

.1660

.1013

.0376

2 ~ Field Values

.0247

.0117

.0050

.0031

.0010

.0006

.0061

.0269

.0720

.1384

.2280

.3274

.4548

.6356

.8398 1.0846

1.4040 1.7859 2.1371 2.3322 2.4580

2.4068 2.2611 2.0127 1.6509 1.2987

.9459

.6471

.4196

.2717

.1652

.0918

.0538

.0249 ,0151 ,0054

.0036

.0016

.0006

.0005

.0002

.000I

COLOR-MATCHING FUNCTIONS 15

total minimum value of the cumulative error [condition (1)] for the 36 points was .0255. The per cent differences between the ratios of the two pairs of func- tions are shown in Table I, and significantly exceed the error in the data only between 460-500 m F and 730-750 m F. The error in the deep reds is most likely due to experimental error, because the aio/Yio ratio becomes constant, while ao/y2 varies inconsistently in this region. The discrepancies between 460-500 m/~ may be partially due to errors resulting from the macular pigment effects. In addition, this is a region where the 10 ~ color-matching functions have been found unsatisfactory (Stiles and Wysecki, 1962), which would imply further sources of errors. These errors are more clearly shown by plotting the per cent differences between the aio/Yio and ao./y2 ratios (Fig. 1). Above 620

W (J Z bJ n- hl LL b_ a

Z I,I 0 rr hl Q.

+40Ix 1 +20

:~_'X\ 0

1

- 2 0 - - 1

- 4 0 4OO

Figure 1.

I I 1 I i I I 1 I I 1 1 I I I I

<%X~x

\X.. X~ f"X"X"x.. X __X,_x.x, x'i /

I I 1 I 1 I 1 I 1 I I I I I • 4 4 0 4 8 0 5 2 0 5 6 0 6 0 0 6 4 0 6 8 0 7 2 0

WAVELENGTH, my. Per cent difference of function ratios for 2 ~ and 10 ~ fields versus wave length.

O r d i n a t e o q u a l s I 0 0 - ~ \ylo y 2 / l \

mF, the error pattern appears random. However, below 620 mF there are obvious nonrandom differences in the results, which suggests a possible correla- tion between the 2 ~ and 10 ~ field color-matching functions.

I f the macular pigment is the principal variable causing differences between the measured 2 ~ versus 10 ~ field color-matching data, then a plot of a2]aio or Y2/7io should be shaped similar to the macular pigment absorption curve. This correlation is only partially obtained (Fig. 2), however, due primarily to the large experimental errors in the violet region, which probably result from variations in the lens transmittance. Nevertheless, the agreement is within

16 WHITMAN RICHARDS

LIJ z:) ._J

>

o m

I--- rr"

1 . 6

i . 4

1.2

1.0

0 . 8

0 . 6

0 . 4

0 . 2

0 4OO

I' I [ I I .... I I I I I I I 1 I ' I "1 / /

- - -

' - _

- - - - 5 - - - M A C U L A R P I G M E N T - -

- - T R A N S M I T T A N C E - -

I I I I I I I I I I I I I I I 1 I 4 4 0 4 8 0 5 2 0 5 6 0 6 0 0 6 4 0 6 8 0 7 2 0

W A V E L E N G T H , m F

Figure 2. Ra t io of cor responding calcula ted co lor -match ing func t ions for the 2 ~ and 10 ~ fields versus wave length, c o m p a r e d w i t h t he macu la r p i g m e n t t r a n s m i t t a n c e

experimental error, and discrepancies occur in the same direction as the consist- ent errors indicated by a comparison of alo/71o versus a2/72. Therefore, there is justification for continuing this analysis on the assumption that the maeular pigment is the principal variable causing differences between corresponding " t rue" color-matching functions for the 2 ~ and 10 ~ fields, for example, between

a 2 and al0, or 7~ and 7~o. Let ao and 70 represent the 2 ~ field color-matching functions after correcting

for ocular media and maeular pigment filtering effects for a 2 ~ field. I f t is the lens and ocular media transmittance, ~ is the macular pigment transmittance, and k2 is a fraction (independent of wave length) representing the average effect of the maeular pigment on color-matching functions for a 2 ~ field, then:

o" 2 (6) o" 0

t [1 - (1 - ~ ) k ~ ]

~ (7 ) yo = t[1 - (1 - ~ ) k 2 ]

However, if the primary difference between corresponding color-matching func- tions for the 2 ~ and 10 ~ fields is the difference in the relative effects of the macu- lar pigment, then:

(710

a~ ~ t[I -- (1 -- F)/Clo] (8)

71o 70 ~ t[1 - - (1 - - ~ ) k ~ o ] (9)

COLOR-MATCHING FUNCTIONS 17

for the appropriate value of klo. In this case, k~o is a fraction (independent of wave length) representing the average effect of the macular pigment on color- matching functions for a 10 ~ field. Therefore, for the proper cboice of k2 and klo:

a2 (YlO (10) t [1 - (1 - /x) k 2 ] t [1 - (1 - / x ) k l o ]

Y2 Ylo X (11) t [ 1 - ( 1 - g ) k2] t [ 1 - ( 1 - g ) k~o]

where the approximation is valid within the limits that color remains invarient with the field size.

Fortunately, there is only one pair of k~. and klo values which will best satisfy equations (10) and (11). The uniqueness of these values can be proven by assuming that another pair of values kx and k~ will also satisfy equations (10) and (11). I f the approximations of equations (10) and (11) are equally accurate for both k2, klo and kx, k~, then:

1 - - (1 - - /X) klO = 1 - - (1 - / ~ ) k y ( 1 2 )

1 - (1 - /x) k 2 1 - (1 - /L) k x

and k2 - k l o = kx - k y + (1 - ~ ) ( k 2 k y - k~okx) .

However, because (1 , ~) is an independent variable

k 2 k ~ - k l o k ~ = 0

and

(13)

(14)

kx k2 k-~ = klo ( is)

The remaining terms of equation (13) give

k 2 - k l o = k z - k u (16)

which, together with equation (15), requires that

k 2 = kz , k l o = ky , (17)

providing that k 2 # klo. Therefore, k 2 and klo are a unique pair of values. In order to determine k 2 and kl0, values for both sides of equations (10) and

(11) are calculated at 36 equally spaced wave lengths for many possible values of k 2 and ]Qo. The desired values of ]c 2 and klo are those two values which most closely equated both sides of equations (10) and (11) at the same time, over all wave lengths. The "best-fit" trial-and-error solution with t and /z taken as shown in Table I gives k 2 = .45 and klo = .14. These values correspond to a

2--B.~.B.

18 WHITMAN !RICHARDS

minimum effective maeular pigment transmittance (at 460 m/x) of .63 and .88 for the 2 ~ and l0 ~ fields, respectively. The resultant values of a 0 and 7o ealeu- lated from equations (6) and (7) for the 2 ~ field data are given in Table III . These values should be eonsidered only as indicative of the true action spectra, beeause of the crudeness and of the approximations of the above eoneept of the maeular pigment effects.

The shape of % clearly indicates the presence of at least two components. The simplest assumption, tha t g0 or a2 is composed of only two components, requires tha t the ~ and ~ terms of a~ be grouped to represent the long-wave component where 5 = 0. This component is designated as a~.. Therefore, the short-wave component of a2, designated as flu, must be proportional to 5 with the coefficients of ~ and ~ equal to zero. I f this were not the case, then any contribution of ~ or ~ to f12 would cause secondary peaks in the fl~. component. Therefore, since a 2 = a u + fi~ and fl~. = c15, then

a2 ---- au -- f~2 = 1.50~ + 1.09~ + .075 -- c15 (18) and

a 2 = 1.50~ + 1.09~ + c25 (19)

where c 1 + c2 = .07. The constant c u can be determined from tritanopic color- matching functions, if it is assumed tha t the blue, or flu, mechanism is absent.

In this ease, the tritanopic color-matching functions will be a2 and ~2, and the ratio au/Tu will be the same at each pair of wave lengths confused by tritanopes. Let a'2/7' 2 and a~/~,~ be the ratios of the true color-matching functions a2, Y2 at two equivalent wave lengths. Then a measure of how closely the function a 2 satisfies the tritanopic color-matching data is:

where e is the experimental error. Therefore, the best value of % is the one which minimizes condition (20). Appropriate values for e are taken from Table I such that:

e 2 = E~ + E22, (21)

where E~ and Eu are the experimental errors for the color-matching functions at the equivalent wave lengths. By using Wright and Thompson's (1953) tri- tanopie equivalent wave lengths of 550-412, 540-416, 530-421, 520-425, 510- 429, 500-434, 490-439, 480-445 m/x, the constant % is found by iterative approximations to be - .29 . Therefore,

a 2 = 1.50~ + 1.09~ - .295 (22) and

f12 = .365. (23)

COLOR-MATCHING FUNCTIONS 19

T A B L E I I I

Values for the normal ized " t r u e " color-matching funct ions calculated f rom the stand- ard 2 ~ field color-matching functions. These functions are corrected for ocular media

and macular p igment filtering effects.

Wave length m/~

400 410 420 430 440

450 460 470 480 490

500 510 520 530 540

550 560 570 580 590

600 610 620 630 640

650 660 670 680 690

700 710 720 730 740 750

Er 0

.3835

.9863 2.1637 3.1334 3.0790

2.6895 2.2716 1.6161 1.1655

.9607

.9802 1.2378 1.7161 2.2253 2.6699

~o

.0656

.1702

.3891

.6418

.7872

.9298 1.1448 1,3332 1.7174 2.1897

2.6544 3.3168 4.1138 4.5996 4.7568

(Z 0

.0316

.0780

.1614

.2146

.1878

.1277

.0644

.0633

.1959

.4371

.7266 1.1148 1.6608 2.1969 2.6567

3.0869 3.4730 3.7598 3.9302 3.9478

3.7277 3.3154 2.6892 1.9539 1.3291

.8248

.4727

.2455

.1286

.0616

.0306

.0155

.0077

.0037

.0019

.0008

4.6890 4.4002 3.8601 3.1609 2.4039

1.6877 1.1259

.7147

.4285

.2499

.1391

.0744

.0373

.0190

.0089

.0043

.0023

.0009

.0005

.0004

.O001

3.0813 3.4705 3.7585 3.9292 3.9471

3.7272 3.3152 2.6891 1.9539 1.3291

.8248

.4727

.2455

.1286

.0616

.0306

.0155

.0077

.0037

.0019

.0008

.3519

.9083 2.0023 2.9188 2.8912

2.5618 2.2072 1.5528 .9696 .5236

.2536

.1230

.0553

.0284

.0132

.0656

.0025

.0013

.0010

.0007

.0005

.0002

.0001

.0000

.0000

.0000

.0000

.0000

.0000

.0000

.0000

.0000

.0000

.0000

.0000

.0000

20 WHITMAN RICHARDS

The values of a 0 and fl0 determined in the same manner as a 0 and Yo from equa- tions (6) and (7) are given in Table I I I .

A plot of a o and ~o versus frequency shows a secondary hump in each curve near 440 m/z. Because these humps appear at the same wave length as the maximum of rio, they imply that a fraction of/30 may also be a component of a 0 and 7o. The hump in 70 can be eliminated by subtracting (1/6)fl o from ~o, giving a smooth curve. Similarly, a o can be smoothed by subtracting (1/15)fl o, providing the a o values are modified slightly in the region 440-500 m/z. These changes are largest at 470 and 480 m/z, as indicated in Figure 3. The modified values of %, 70, and of their two possible components are given in Table IV.

The fl component of the a and 7 action spectra is believed to be related to the differences between the 2 ~ and 10 ~ field response functions. I f the receptor responsible for the a (or y) action spectra contains two pigments, ~ and fl (or 0 and fl), then changes in the relative concentration of fl will change the a (or ~) action spectra. The action spectrum produced by a new fl concentration would not be linearly related to the original action spectrum, because an action spectrum resulting from a mixture of pigments is a nonlinear function of the relative amounts of its components. The dip at 470 m/z in the unsmoothed a curve in Figure 3 probably represents such a nonlinear addition of two pigments. In a similar manner, the irregularities in Figures 1 and 2 can be partially ex- plained by postulating differences in the fl concentration between the 2 ~ and 10 ~ fields.

A second effect of a fl component is possible if the photosensitive molecules are arranged in a semi-ordered array in the receptor. Such an arrangement might permit dipole-type exciton interactions between neighboring pigment molecules (Kasha, 1959). These interactions would raise (or lower) the radiant energy threshold of the photosensitive molecules, which would then require more (or less) energetic quanta before absorption takes place. The net effect would be to cause a shift in the absorption spectra along the frequency axis toward higher or lower frequencies, depending upon whether the spacing between the molecules increased or decreased, respectively. The effect could be enhanced by a suit- able impurity, perhaps such as the photosensitive fi component in the above action spectra. I f the concentration of the fl component increases with increas- ing distance from the fovea, then the action spectra for extraofoveal positions would be shifted toward the blue, as is the case for the ~10 response function, when compared with the ~2 response function. (See Table I I and Fig. 4.) Considering that the method of analysis would minimize and tend to obscure any real effect of this nature, the shift may be significant.

COLOR-MATCHInG FUNCTIONS 21

T A B L E I V

Values for t h e modi f ied , n o r m a l i z e d t r u e c o l o r - m a t c h i n g f u n c t i o n s a n d for t he i r poss ible c o m p o n e n t s . These f u n c t i o n s r e p r e s e n t a c t i o n spec t r a for 2 ~ fields, a n d a re

i n d e p e n d e n t o f ocu la r m e d i a a n d m a c u l a r p i g m e n t f i l te r ing effects.

Wave length m/~

400 410 420 430 440

450 460 470 480 490

500 510 520 530 540

550 560 570 580 590

600 610 620 630 640

650 660 670 680 690

700 710 720 730 740 750

O~ 0

.0235

.0606

.1338

.1962

.1987

.1894

.1965

.2176

.3033

.4775

.7469 1.1380 1.6671 2.1300 2.6494

3.0813 3.4706 3.7585 3.9292 3.9471

3.7272 3.3152 2.6891 1.9539 1.3291

flo

.3519

.9083 2.0023 2.9188 2.8912

2.5618 2.2072 1.5528

.9696

.5236

.2536

.1230

.0553

.0284

.0132

.0056

.0025

.0013

.0010

.0007

.0005

.0002

.0001

.0000

.0000

.0656

.1702

.3891

.6418

.7872

.9298 1.1448 1.3332 1.7174 2.1897

2.6544 3.3168 4.1138 4.5996 4.7568

4.6890 4.4002 3.8601 3.1609 2.4039

1.6877 1.1259

.7147

.4285

.2499

$o = ao - 3o/15

.0000

.0001

.0003

.0016

.0059

.0186

.0493

.1140

.2377

.4426

.7300 1.1298 1.6634 2.1281 2.6485

3,0810 3.4705 3.7585 3.9292 3.9471

3.7272 3.3152 2.6891 1.9539 1.3291

.8248 ,4727 .2455 .1286 .0616

.0305

.0155

.0077

.0037

.0019

.0008

.0000

.0000

.0000

.0000

.0000

.0000

.0000

.0000

.0000

.0000

.0000

.1391

.0744

.0373

.0190

.0089

.0043

.0023

.0009

.0005

.0004

.0001

.8248

.4727

.2455

.1286

.0616

.0306

.0155

.0077

.0037

.0019

.0008

Oo = ~ o - f lo /O

.0069

.0188

.0554

.1553

.3053

.5028

.7769 1.0744 1.5558 2.1024

2.6121 3.2963 4.1046 4.5949 4.7546

4.6881 4.3998 3.8599 3.1607 2.4038

1.6876 1.1259

.7147

.4285

.2499

.1391

.0744

.0373

.0190

.0089

.0043

.0023

.0009

.0005

.0004

.0001

22 WHITMAN RICttARDS

I l l U3 Z 0 s

la.I nr"

UJ >

I . --

_1 I.U n"

WAVELENGTH, rnff.

400 440 480 520 560 600 640 720 6 . o o ~ I m i f I r m m m l ~m ~ J t ~

._1

4.00 ~ ' o ~ -4

2.00

1.00 0.80 0,60

0.40

0.20

0 . 1 0 ~ 0.08 0.06 0.04

0.02

/

0.01

/

',,

\ /

- / / . / / / /

24 22 20 18 16 14

WAVENUMBER 10-Sx cm -I

Figure 3. Semi-log plot of relative response versus frequency of the normalized " t rue" color-matching functions ao, ~o, f~0, and of their proposed components. The solid lines indicate the "true" response functions or "action spectra"; and the f~o component ofao and yo is also indicated by solid lines. The long dashed lines indicate the postulated ~bo and 8o components of ao and 7o. The dotted line indicates the calculated values of ao before smoothing, All functions are for 2 ~ field stimuli and are independent of ocular media and

maeular pigment filtering effects

COLOR-MATCHING FUNCTIONS 23

la.I O3 Z 0 13. r

c f

la.I >

i.'-- < ._1 W n"

400

6 . 0 0 - -

4 . 0 0 - -

2 . 0 0 - -

1.00 --- 0 . 8 0 - - 0 . 6 0 --

0.40~

0 . 2 0

0 . 1 0 0 . 0 8

0 . 0 6

0 . 0 4

4 4 0 I !

O [ o f / \

WAVELENGTH, m/z

4 8 0 5 2 0 5 6 0 6 0 0 6 8 0 I I I I I I I I I I I I

/ ,,\ / / . / ,,,

11

oo i// 0 . 0 1 I

24 E t I I I 1 t

2 2 20 18 16

WAVENUMBER 1 0 - 3 x cm "1

14

F i g u r e 4. Semi-log p lo t o f r e l a t ive response ve r sus f r e q u e n c y of t h e ca l cu la t ed q a n d y va lues for b o t h t h e 2 ~ a n d 10 ~ fields, before co r rec t ing for ocular m e d i a a n d m a c u l a r p i g m e n t effects. The differences in m a c u l a r p i g m e n t effects a re n o t s igni f icaut for wave n u m b e r s be low 19,000 c m - 1 , a n d the re fo re t h e d i s p l a c e m e n t in t h e ylo ve r sus Y2 p e a k is s ignif icant . The solid l ines i nd i ca t e 2 ~ field d a t a a n d t h e d a s h e d l ines i nd i ca t e 10 ~

field d a t a

24 WHITMAI~ RICHARDS

One other theoret ical implicat ion of the present analysis is t h a t the a func- t ion appears to represent the l inear summat ion of a "b lue" (fl) and a " r e d " (a) response funct ion. This appa ren t summat ion could be an ar t i fact of the anal-

ysis only i f the addition of fl to a minimizes the differences between alo/71o and a2172 in the "b lue" region where fl can provide a significant cont r ibut ion to a. However , because a is less t han 7 in this region, any such differences here can best be minimized b y subtracting fl f rom a in order to make the a/7 rat ios as small as possible. Therefore, if the a2 and al0 values are in error, i t would be in the direct ion of too l i t t le ft. Ano the r confirmation of the real i ty of the a funct ion is given in Figure 1. This figure shows t h a t the present a2 and ale funct ions now balance the per cent difference between a2/72 and alo/71o in the region where fl would have its m a x i m u m effect (near 440 m/~). A n y fur ther changes in the a moun t offl , e i ther b y addi t ion or b y subtract ion, would increase the value of condit ion (1). These facts suggest t h a t the a funct ion m a y repre- sent one response component of the visual process.

The use of the Bendix compute r was made possible th rough the cour tesy of Dr. G. E. Fischer a t the Cambridge Elec t ron Accelerator, Cambridge, Mass.

During the prepara t ion of this repor t aid was received under grant 3TIGM- 1064-02S 1 f rom the Nat ional Ins t i tu te of General Medical Sciences to the M.I.T. Psychology Section.

L ITERATURE

Judd, D . B . 1951. "Basic Correlates of the Visual Stimulus." Handbook of Experi- mental Psychology, S. S. Stevens, ed., New York: Wiley.

Kasha, M. 1959. "Relation Between Exciton Bands and Conduction Bands in Molecu- lar Lamellar Systems." Biophysical Science--A Stud~j Program, J. L. Oncley, ed., New York: Wiley.

Stiles, W. S. 1958. "The Average Colour Matching Functions for a Large Matching Field." Visual Problems of Colour, N P L Symposium ~u 8, London: Her Majesty's Stationery Office.

Stiles, W. S. and J. M. Bureh. 1959. "NPL Colour Matching Investigations." Optica Acta, 6, no. 1, p. 1.

Stiles, W. S. and G. Wysecki. 1962. "Field Trials of Color Mixture Functions." Jour. Opt. Soc. Amer., 52, p. 58.

Wright, W. D. and L. C. Thompson. 1953. "Convergence of Tritanopie Confusion Loci and Derivation of the Fundamental Response Function." Jour. Opt. Soc. Amer., 43, p. 890.

]~,ECEIVED 2-25-62