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Luminance matching functions
Vision Res.Vol.lO,pp.563473.PergamonPrcu 1970.Printed inGrut Britain.
LUMINANCE
MATCHING D.A.
FUNCTIONS
PALMER'
National Physical Laboratory,
Teddington,
Middlesex
(Received 31 December 1969)
INTRODUCTION EXPEIUMENT has shown (PALMER,1966,1967,1968) that the equivalent luminance of any light at any level may be expressed to a good approximation as a function of its scotopic and photopic luminances. This principle holds without exception for the 19 persons whose vision has so far been tested. A related theorem is that lights that match each other at high and low levels will also match at intermediate levels, if the attenuation is spectrally neutral (PALMER, 1966). Conversely, a luminosity match which is stable through the mesopic range must be matched in scotopic and photopic luminance. In order to match a given test stimulus in this manner, a mixture of at least two different instrumental stimuli is required, in the proportion that produces the same scotopic luminance in both halves of the field, and similarly for the photopic luminance (PALMER,1966). The relationship between the stimuli for such a match may be expressed mathematically in a manner analogous to the colour matching equations for a dichromat e.g.,
GI (G) = Yt (Y) + 4 (B).
(1)
This equation means that for a stable luminosity match between a light (G) and a mixture of (Y) and (B), the stimuli must be in the ratios G1: Y,: Bi. Since in this connection the ratios are more significant than the absolute quantities, the equation may be normalised for a unit amount of the test stimulus: (G)=y,(Y)+b,(B)
(2)
where yI = Y,/G,, etc. For a monochromatic
light C(X) of wavelength X the equation becomes: (G(h)) = 71 (4 (r> + 6, (A) (B)
(3)
where j$ (X) and 6, (X) are analogues to the colour matching functions. By semantic analogy, they may be termed luminance matching functions. Various systems of units could be employed in such equations. In this account, each stimulus has been measured as a radiant flux, so that the units are based on measurements of physical power. ‘Present address: Institute of Ophthalmology,
Judd Street, London W.C.1. 563
564
D. A.
PALMER
The symbols Y, G, B etc., are convenient to represent yellow, green and blue lights respectively, although coIour appearance is not strictly relevant to the present task of Iuminosi~ matc~ng. The notation brings out the close analogy with colour matching functions, with which, neverthelesss, the luminance matching functions should not be confused. The particular stimuli in the equations are to be taken merely as examples. This paper will present some investigations of the luminance matching functions, in particular how they may be measured. Firstly some characteristics will be deduced from an expected relationship to V’(A) and V(X). Equation (2) is postulated to hold for matches at all Ievels. At low levels therefore the scotopic luminances must be equal, that is: v(a)
= ;((a) y:
+
6, (a) V’b.
Similarly at high levels, the photopic luminances must be equal: F(h) = Jr 0) v, t 6, (3 v,.
(5)
Since these are equation of luminance, the equality sign replaces the equivalence sign. I”,,, V’,.: Vb, V, are the values of v’(h) and Y(X)at the wavelengths of the instrumental stimuli here supposed monochromatic. For non-monochromatic stimuli there are corresponding constants of the form : p, = I#F (4 WV dVf#y
69 dA
G-3
where +y(X)is the spectral power distribution of the yellow stimulus, with similar expressions for the blue stimulus and for the scotopic values. The solutions of (4) and (5) are:
In words, the luminance matching functions are linear transforms of V’(A) and V(A). Therefore, provided that V’(X)and V(X)are themselves sufficiently additive, it wouId appear that the iuminance matching functions may be manipulated algebraically as if they were coiour matching functions. WRIGHT(1969) describes the calorimetric analogues of this procedure, which was to be tested in the present experiments. METHODS Apparatus
The apparatus was Stiles’ Trichromator at N.P.L. (STILES,1955). The main modification from Stiles’ conditions was the enlargement of the angular subtense of the circular field from 10’ to 1.5”.Otherwise it was similarly divided symmetrically horizontally, the test stimuli occupying the upper half. The circular surround, which was physically part of the lower field (Sruss, 195.5),was also enlarged proportionately to a diameter of 20”. For a few experiments with extrafoveal vision the subtense of the field was masked down to 5’ (circular, divided symmetrically horizontally, with a dark surround). A fixation spot 20’ in subtense appeared at an angle of either 7’ or 14’ from the centre of the field on the temporal side. (The right eye was used throughout.) This spot had the spectral composition and i&minance of the Iower half of the field. The instrumental stimuli were at 470.6 nm (B), and 558-2 nm (Y), as chosen by STILES(1957). Other detaiis such as the band-pass of the inept were exacdy as STUSS(1955) describes. The radiant fluxes were measured directly after each match with a calibrated photomultiplier, as described by CRAWFORD (1965). The linearity and stability of this type of detector are not above suspicion (CRA\KFORD, 1962). However, the present cell, which had been used in various tests of additivity in photometry (PALMER, 1966) and calorimetry (LOZANO and PALMER,1967), invariably proved linear to within + 2 per cent over a wide range of iltuminances.
Luminance Matching Functions
565
Measuring luminance matchingfimctions The following method was developed to find the particular mixture of yellow and blue stimuli which would match a given test stimulus in luminosity at all levels. The test stimulus was set at a high level and mixtures of yellow and blue in various proportions were matched to it in luminosity by direct comparison, with central fixation. The last proved unimportant (see later). The instrument controls were ganged together (Srnm, 1955) so that matches were easily made without altering the proportions in the mixture. After each match the setting was disturbed at random, and matches were repeated until three or four in good agreement were obtained in succession. The arithmetic means of the accepted readings were calculated. The proportion of blue stimulus required to match a unit amount of test stimulus was plotted against the corresponding quantity for yellow. The resultant graph was generally a curve, ranging from all-yellow, no-blue to all-blue, no-yellow, as in Figs. 1,4 and 5 which will be described later. This procedure was repeated at successively lower levels, with allowance for progressive dark-adaptation, to produce a family of curves with various slopes related to the Purkinje shift. According to the hypothesis, these curves should intersect at a unique point, which should indicate the co-ordinates (y,, b,) for equivalence, in terms of the experiment, with the test stimulus. For monochromatic test stimuli, these co-ordinates are the spectral stimulus values, the set of which constitutes the luminance matching functions. Curves could perhaps be fitted mathematically to the data. However, the theoretical form of the curves is as yet unknown (and indeed TESTIERand BLOTTIAU[1951] found several varieties), so the intersection point was estimated by eye from freehand drawing. The justification for this procedure will be evident when the graphs are discussed. The analogy with the colour matching functions suggests that a red or orange test stimulus will require “desaturation” with blue before the match can be made. In such a case, the yellow instrumental stimulus was set to the required level and matched successively with various mixtures of the test and the blue stimuli. Similarly, violet test stimuli required admixture with the yellow stimulus. Experiments The luminance matching functions were investigated for the author’s right eye, which had normal colour vision and normal photopic and scotopic sensitivities. Detailed measurements were made by the above method for stimuli at 530 nm, 440 nm and 620 mn, the last two requiring admixture, as just described. The numerous illuminance levels extended from 100 to O-01 trolands. Experiments were made at these three wavelengths with the 15” field viewed centrally, and with the 5” field viewed centrally and eccentrically at 7’ and 14’ on the nasal side, by using the small fixation spot already mentioned. The effects of adaptation to blue, green and red lights were cursorily investigated for the above three wavelengths in the 15” field. The adapting light was a flash torch covered with layers of diffusing paper and the appropriate coloured filter. The blue filter was Wratten 47B, with a maximum transmission at 440 nm. the green was Wratten 74, peaking at 540 mn, and the red, Wratten 70. transmitting above 650 nm. This source was held close to the eye for about 1 min. The illuminance was about 20 times that of the instrument field. After adaptation, the match was made as quickly as possible, with readaptation before the next match. With this variation, the procedure was as before. Spectral wavelengths from 380 nm to 740 nm at 30 run intervals were matched by the basic technique but
only in the 15”field, and sometimes to save time, at only two levels, namely 1 troland and @03trolands. The points of intersection of the resultant graphs were estimated by eye to provide the stimulus values for the luminance matching functions. The functions were also measured for some wavelengths in the 15” field, by an analogue of Maxwell’s method for colour matching (CRAWFORD, 1965). Mixtures of the instrumental stimuli in the test field were 6rst matched (in terms of the experiment) to a standard white (w) in the mixture field. This was N.P.L. Bluish White (Srr~ss. 1955). Then the soectral stimuli were simihuly matched to the standard by adding to each either the yellow or the blue stimulus as
appropriate. The equations are analogues of those described by CRAWFORD (1965):
m = Y’I (Y)
+ b’, 03)
whence
so that In practice either y, (I\) or b,(A) was zero, only the appropriate
stimulus being added to the test stimulus.
566
D. A. PALMER
Four non-mon~~omatic lights, which were those numbered 4,5,7,8 in a previous publication (Pw, 1967), were aiso matched by the technique in the 15”field by usingnumerous ili~~~~levels. The analogues to the tristirnuius values in cotorirnetry, which wiIl be termed stimulus values in this account, were found both by the direct method of observation and by the Maxwell-type of method, for comparison. To test additivity, they were also calculated as the integrals of the products of the luminance matching functions and the spectral power distributions. The last were measured by running a slit through the spectrum of the instrument whilst recording on the photomultiplier (PALMER,1966). If additivity holds, the mathematically predicted values should agree with the observed. RESULTS
The results appear in the figures and the table. Figure 1 shows the proportions of yellow and blue ins~ument~ stimuli in various mixtures which matched in luminosity a monochomatic stimulus at 530 nm in the 15” field, centrally viewed. The quantities (yr, fi,) are normalised for unit power of the test stimulus. Each graph was obtained at the ilfuminance indicated
PK.?.I. Mixtures CylbJ which matched a unit amount of the stimulus at 530 nm in the 15”fieId at various illuminances indicated at the feet of the respective graphs.
at its foot. It can be seen that the precision of observation is good; variations greater than f 10 per cent between matches were rarely found. Figure 2 is simiIar to Fig. 1 except that the monochromatic test stimulus is at 440 nm. The yellow stimulus was added to the test, and is shown conventionally as a negative quantity. Figure 3 is as Fig. 2, but for the test stimulus at 620 nm. The blue stimulus is now added to the test. Figure 4 is as Fig. 1 but for the 5” field at the peripheral angles 0”, 7’ and 14”, as indicated beside the respective curve. The points of inflection for the three conditions of observation are indicated by stars. These rest&s are for the 530 nm stimulus. Corresponding experiments
567
Luminance Matching Functions
C
-3
-I
-2
I
0
YI FIGS.2. As Fig. 1, but for the stimulus at 440 MI. Note the negative values for y,.
o-
-0.1
-
-0.2
-
bl
-0.3 l n ”
^C
V’3
I’0
_
YI FIG. 3. AS Fig. 1, but for the stimulus at 620 run. Note the negative values for b,.
D. A. PALMER
FIG. 4, As Fig, 1, but for the 5’ field at various peripheral angles which, together with the are indkated at the foot of each graph. The intersections for the three peripheral m&s are starred, and should be compared with that in Fig. 1.
illutiances,
0
I
2
3
4
5
6
7
3
9
IO
YI
FE. 5. As Fig, 1, but for mat&es after adaptation to blue, green and red IigMs @, g, rl. 3%~ interstctions for the three umditims of adaptation are star&, and &mid be cumparcd with that in Fig. 1.
569
Luminance Matching Functions
A.
nm
Fro. 6. Luminan Ce matching functions measured by the dixect method (solid points) aad the MaxweWpemethod (open points). n , yr(A); 0,6&l). The vertical lines indicate the instrumental stimuli. I.0 -
0.1
V 0.01
.
0.001
\
L
1 400
I 600
I 500
A.
p(:)
\ . \\ \ ‘r I
700
nm
tineai’ transforms of the huninance matching function compared with v’ (A) (---), and J’, (A)(- - - -) (lower curve). ., 096 1, (A) + 0.6 6, (A);& 0.7 g,(A) : 0-z 61 (A); n , O-7YI (A) + 0% (A). The last two are indistinguishable above 600 MI, where also VI0 (A) sc V, (A). VJt. IO/74
5-m
D, A. PA.WER
with the 440 nm and 620 nm stimuli, which are not iflusfrated here, displayed a &m&r slight variation of the point of intersection with field size and peripheral angle. Figure 5 illustrates the effect of adaptation to blue, green and red lights, for matches wit& 530 nm stimulus in the 15” field, centrally viewed, as for Fig. f . The intersections for the three sets of graphs are indicated by stars. Corresponding experiments with stimuli at 440 nm and 620 nm displayed a similar slight variation of the point af intersection with adaptation. Figure 6 shows the Iumiuance rn~~~~~~~functions measured in the f 5” field by the direct method and the Naxweif-type method. Figure 7 shows various linear transforms of these functions compared with some C.I.E. V’(X)and V(X)curves. The Table contains the stimulus values for the four non-rnonoch~~t~~ Iights. The values calculated from the “direct’” luminance matching functions and the spectrai power distributions are compared with those observed by the direct method and by the Maxwefttype method. DHGUSSION
The very distinct points of intersection of the graphs in Figs. i-3, which are only a selection from similar results for some 20 lights, show that the proposed stable luminosity matches may indeed be made, at least by one observer. For the three fights tested in detail, the matches were almost unaffected by field size. A very sfight, and probably insignificant variation in the point of intersection with per~pherai angle is shown in Fig. 4, which shotid be compared with Fig. 1, for the $5” field. These conclusions were borne out by the visual appearance of the fieid in the mesopic range. A partial central scotoma usually appeared in the side of the field with the greater scotopic luminance. The scotama disappeared when the field was matched both in scotopic and photopic hnninance, so that a ~~rn~~~s~~~ match was then ~tab~~~~ed ah along the dividing line. Such a match is obviously hardly affected by field size or peripheral angle, a least within the ranges examined here, and hence the fixation need not be carefully controlled. Adaptation had a larger, and probably more significant effect (Fig. S), which was not investigated sufEcient.ly for other tonelusions to be drawn. It is futile to attempt to measure l~~nan~e matching functions with a small field centrally viewed, because there is then no Purkinje shift. Nevertheless, the C.I.E. V,(h) curve for a 2” SeId can also be regarded as a transform of these functions, and thus part of the same scheme (Fig. 7). This coincidence is closely related to the apparent invariance of the Iuminance matching functions with respect to field size and peripheral angle. Another related manifestation is that V&X) happens to be a linear transform of Y’(X) and V, (X) fPkU.XInR,1868). The Maxwell-type of measurement produced results very sim.iIar to those of the direct method (Fig. 6) except that the Maxwell ji, (A) function is significantly the larger in the region between the stimuli. The general agreement between the methods suggests that the quantities in the matches may be manipulated ~~braica~y to predict other matches made with different stimuh. fn further support of this conch&on, hnear transforms of the tuminante functions can be found (Fig. 7) which cXosefyresemble Y’(h) and Y(h), That these pure rod and cone responses can be derived from mesopic data greatly supports the theory. The dire& tests of addit~~ty with no~~rno~~hrornati~ rights ffable f ) display s~~~~~nt differences between the stimulus values found by the three methods, that is by calculation
Luminance Matching Functions
571
from the “direct” functions, and by “direct” and Maxwell-type observations. The “Maxwell” and the blue stimulus values are particularly in disagreement. It should be noted that the largest percentage discrepancies occur in the smallest stimulus values e.g., in B for lights 7 and 8, and thus are not of great significance. Although the luminance matching functions are not additive within observational precision (generally about &lo%, as already mentioned), they are probably little worse in TABLE 1
Stimulusvalues
Test light
Observed
Calculated
4 (Green) 5 (white) 7 (Purple) 8 (Yellow)
Y 0.539 B 0.726 Y 0.491 B O-295 Y 0.170 BOG45 Y 0.610 BO-099
Observed discrepancies from calculation (“A
Direct
Maxwell
Direct
Maxwell
056 0.66 0.48 0.27 0.17 0.033 0.56 0.094
0.35 0.81 0.38 0.33 0.14 0.054 0.47 0.14
+4 -9 -2 -8 0 -27
-35 +12 -22 +11 -17 f20 -23 f49
this respect than their analogues, the large-field colour matching functions (LOZAKO and PALMER,1967, 1968), and the V(h) curve (PALMER,1968). On the other hand, the required match once established in given field conditions is unaffected, within the precision of observation, by changes of illuminance. The curvature of the photopic graphs (e.g. Fig. 1) is clearly the consequence of photometric non-additivity. Nevertheless, the curves all intersect precisely at the point defined by their more linear scotopic counterparts. This result obtains even with large colour differences. Such observations suggest a general theorem. Three or more stimuli may be arranged to produce two spectrally different lights whose luminosities will match by direct comparison, within the precision of observation, at all levels. For three stimuli the match is unique. (Neutral attenuation is assumed, and field size, fixation and adaptation should be controlled as they may have slight effects.) A generalisation drawn by induction from one person’s observations of a limited range of test lights must be tentative. However, a corollary has been widely tested. In previous experiments (PALMER,19661968), 19 individuals had matched in luminosity several nonmonochromatic stimuli to the range of spectral stimuli at various illuminances. Each non-monochromatic stimulus could always be matched within observational precision by a particular monochromatic stimulus in a characteristic ratio, whatever the level. Experiment thus verifies a special case of the theorem, whose conditions are fully met, since any nonmonochromatic stimulus is composed of at least two stimuli. (The proportions of the components cannot be altered without changing the stimulus, so the stable match can be made only with a particular spectral stimulus.) The near-additivity of this type of match and its partial invariance with respect to adaptation obviously suggest Grassmann’s Laws in calorimetry (WRIGHT, 1969), which could perhaps be reframed to include such stable luminosity matches. Since V(X)can serve
572
D.A.
PALMER
as one of the colour matching functions, perhaps only V’(h) need be added to the colour matching functions to produce a set of four functions which, for the normal observer, might be necessary and sufficient to define matches stable in colour and luminosity over a wide range of illuminances. The suggestion remains speculative until tested by experiment. The visual receptors might change their responses to colour, whilst maintaining their response to luminance. The suggested matches might then break down in colour at various levels. Such discrepancies have been found in corresponding experiments with a deuteranope (PALMER,1969). To conclude with considerations for practical photometry, it appears unnecessary to derive the fundamental curves from measurements in pure scotopic and photopic conditions. Luminance matching functions can be obtained from mesopic measurements, with little attention to field size and peripheral angle. Standard photocells need not be calibrated exactly with respect to defined v’(h) and Vo\) curves. Any linear transforms of the luminance matching functions would serve in practice, except those with negative lobes. This freedom might facilitate the design of spectral correction filters (CRAWFORD,1962). To complete this scheme, the luminance of any test light at any level, under given field conditions, would have to be known as a function of the integrals of the luminance matching functions with respect to the spectral power distribution. A method of deriving this function experimentally has already been outlined (PALMER,1967), and as it does not necessarily require monochromatic test lights, it differs fundamentally from the apparently similar process of determining the so-called mean mesopic V(A)functions (HOUGHand RUDDOCK, 1969). The effect of field size can be allowed for by a parameter in an empirical expression for the luminance (PALMER,1968). The disadvantage of this universal system is that two physical measurements would generally have to be made to determine each luminance, though practice might teach when one would suffice. The luminance matching functions of the average Observer, which are not yet available for discussion, might therefore be applied to practical photometry. The functions of 15 individuals have since been determined, with the intention of publishing them shortly. Acknowfe&emenfs-Dr. B. H. CRAWFORD’Sadvice is gratefully acknowledged. The work described in this paper formed part of the officialresearch programmeof the National Physical Laboratory.
REFERENCES CRAWFORD, B. H. (1962). Physical photometry. N.P.L. Noteson Applied Science No. 29. H.M.S.O., London. CRAWFORD, B. H. (1965). Colour matching and adaptation.
Vision Res. 5, 71-78. HOUGH, E. A. and Rt.mooorr, K. H. (1969). The parafoveal visual response of a tritanope and an interpretation of the Vh sensitivity functions of mesopic vision. Vision Res. 9, 935-946. LQZANO, R. D. and PALMER,D, A. (1967). The additivity of large-field colour matching functions. Vision Res. 7,929-937. L.OZANO,R. D. and PALMER,D. A. (1968). Large-field color matching and adaptation. J. opt. Sot. Am. 58, 1653-1656. PALMER,D. A. (1966). A system of mesopic photometry. Nature, Land. 209,276281. PALMER, D. A. (1967). The definition of a standard observer for mesopic photometry. Vision Res. 7,619-628. PALMER, D. A. (1968). Standard observer for large-field photometry at any level. J. opt. Sot. Am. 58,12961299. PALMER, D. A. (1969). Colour matches which include equality of scotopic luminance. Proc. 1st. Congress A.Z.C., Stockholm, 1969. Musterschmidt VerIag, Gottingen. In press. STILES,W. S. (1955). 18th Thomas Young Oration: the basic data of coIour-matching. Yearbook the Physical Society, pp. 44-65. STILES,W. S. (1958). The average colour-matching functions for a large matching field. Proc. 8th N.P.L. Symp. 1957,~~. 211-247,H.M.S.O.,London.
of
Luminance TESER, M. and B-u, antes photopiques.
F. (1951). Variations
Matching
Functions
des characteristiques
573
photometriques
de l’oeil aux lumin-
Rev. d’Opt.. Paris, 30, 309-322. Wm~trr, W. D. (1969). The Meusurement of Colour, Hilger & Watts, London Abstract-Three or more different stimuli may be arranged in a photometer field to produce two spectrally different lights which match in luminosity at all levels. A monochromatic stimulus may be so matched by a characteristic mixture of two instrumental stimuli, the quantities of which for a unit test stimulus may be defined as the spectral stimulus values. The set of these values constitutes the luminance matching functions, which may be manipulated algebraically in order to predict the approximate stimulus values of any test light of known constitution. Y’(h) and Y(h) may be derived as linear transforms of the functions, from mesopic data. Applications to practical photometry are suggested. R&amGOII peut combiner dans un champ photometrique trois stimuli ou plus pour produire deux lumi&res spectralement diff&rentes qui sont de meme luminosite a tous les niveaux. Un stimulus monochromatique peut Metreainsi &alid avec un melange caractiristique de deux stimuli instrumentaux, dont les quantit&s pour un test stimulus unite peuvent ftre deli&s comme la valeurs de stimulus spectrales. L’ensemble de ces valeurs constitue les fonctions d&alit& de luminance, qui peuvent 6tre manipultes algebriquement pour predire les valeurs approchtcj de stimulus pour toute lumiere test de composition connue. v’(X) et Y(h) peuvent etre obtenus par transformations lin&ires de ces fonctions a partir de don&s m&.opiques. On suggere des applications B la photometric pratique. Zusammcnfassurrg-Drei oder mehr als drei verschiedene Reize konnen so in einem photometrischen Felde dargeboten werden, dass zwei spektral verschiedene Lichter, deren Helligkeiten einander an allen Niveauxgleich sind, erhalten werden konnen. Ein einf&rbiger Reiz kann somit einer kennzeichnenden M&hung zweier Instrumentreize, deren G&se auf Grund einer Reizeinheit als Spektralreiz ausgedrtickt, ist, gleichgesetzt werden. Die Wertgruppe gleicht den Helligkeitsfunktionen, welche algebra&h so gehandhabt werden kiinnen, das der ungef&re Reizwert aller Testlichter, deren Z usammensetzung bekannt ist, vorausgesagt werden kann. Y (I\) und V(A) k6tmen als Lineartransforme dieser Funktionen von mesopischen Ergebnissen abgeleitet werden, Es werden Vorschlige fur die Ausntitzung der Methode in der praktischen Lichtmessung gemacht. Pearome - Tpa mm 6onee p asnmmbtx crn~ym MOQT 6~13 TSLKLI~HMB none &ITOM~T~~, ¶TO IIpOjQIIKpyioT CIIeKTpaJIbIIO paSIE¶IbIe CBeTOBbIe pa3IIpaXtETeJIE, KOTOpble M0I”y-r ~paarni~aT%c~ no ~~KOCTII Ha BceX YpOBIISIX. MOHOXpOMaTIiKeCKIItJCTI~.&+TMOX~T Bt.t~b. TaKEM o6pa3OM, y-paBIieH CO CMeCbIO JIByx EXCrpyMeIITaJIbIIbIX CTIIMyJIOB, ROJIH¶ecrBa K0~0pbIX ALIBIemor TecTOBOrO CTIIMyna MOI-yT 6brrb OIIpeneJIeIKbf KaIC BCrIeIcTpaJIbaoro CTIIM~M. Ha6op T~KIIX B~IRSAII 06pa3yer &rmmrn cpaanemfa nprocrn, KOTOpbtMkI MOXSIO OuepHpoBaTb a~n-e6pamtecrcn, C IIeJIbIO I.IpeACKa3aTb ITpII6JIE3IITelTbII&Ie B crn~yna nr~a mo6oro Tecroaoro cBeTa n3Becrnoro cocra~a. v’ (A) II V(A) ~oryr
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LUMINANCE
MATCHING D.A.
FUNCTIONS
PALMER'
National Physical Laboratory,
Teddington,
Middlesex
(Received 31 December 1969)
INTRODUCTION EXPEIUMENT has shown (PALMER,1966,1967,1968) that the equivalent luminance of any light at any level may be expressed to a good approximation as a function of its scotopic and photopic luminances. This principle holds without exception for the 19 persons whose vision has so far been tested. A related theorem is that lights that match each other at high and low levels will also match at intermediate levels, if the attenuation is spectrally neutral (PALMER, 1966). Conversely, a luminosity match which is stable through the mesopic range must be matched in scotopic and photopic luminance. In order to match a given test stimulus in this manner, a mixture of at least two different instrumental stimuli is required, in the proportion that produces the same scotopic luminance in both halves of the field, and similarly for the photopic luminance (PALMER,1966). The relationship between the stimuli for such a match may be expressed mathematically in a manner analogous to the colour matching equations for a dichromat e.g.,
GI (G) = Yt (Y) + 4 (B).
(1)
This equation means that for a stable luminosity match between a light (G) and a mixture of (Y) and (B), the stimuli must be in the ratios G1: Y,: Bi. Since in this connection the ratios are more significant than the absolute quantities, the equation may be normalised for a unit amount of the test stimulus: (G)=y,(Y)+b,(B)
(2)
where yI = Y,/G,, etc. For a monochromatic
light C(X) of wavelength X the equation becomes: (G(h)) = 71 (4 (r> + 6, (A) (B)
(3)
where j$ (X) and 6, (X) are analogues to the colour matching functions. By semantic analogy, they may be termed luminance matching functions. Various systems of units could be employed in such equations. In this account, each stimulus has been measured as a radiant flux, so that the units are based on measurements of physical power. ‘Present address: Institute of Ophthalmology,
Judd Street, London W.C.1. 563
564
D. A.
PALMER
The symbols Y, G, B etc., are convenient to represent yellow, green and blue lights respectively, although coIour appearance is not strictly relevant to the present task of Iuminosi~ matc~ng. The notation brings out the close analogy with colour matching functions, with which, neverthelesss, the luminance matching functions should not be confused. The particular stimuli in the equations are to be taken merely as examples. This paper will present some investigations of the luminance matching functions, in particular how they may be measured. Firstly some characteristics will be deduced from an expected relationship to V’(A) and V(X). Equation (2) is postulated to hold for matches at all Ievels. At low levels therefore the scotopic luminances must be equal, that is: v(a)
= ;((a) y:
+
6, (a) V’b.
Similarly at high levels, the photopic luminances must be equal: F(h) = Jr 0) v, t 6, (3 v,.
(5)
Since these are equation of luminance, the equality sign replaces the equivalence sign. I”,,, V’,.: Vb, V, are the values of v’(h) and Y(X)at the wavelengths of the instrumental stimuli here supposed monochromatic. For non-monochromatic stimuli there are corresponding constants of the form : p, = I#F (4 WV dVf#y
69 dA
G-3
where +y(X)is the spectral power distribution of the yellow stimulus, with similar expressions for the blue stimulus and for the scotopic values. The solutions of (4) and (5) are:
In words, the luminance matching functions are linear transforms of V’(A) and V(A). Therefore, provided that V’(X)and V(X)are themselves sufficiently additive, it wouId appear that the iuminance matching functions may be manipulated algebraically as if they were coiour matching functions. WRIGHT(1969) describes the calorimetric analogues of this procedure, which was to be tested in the present experiments. METHODS Apparatus
The apparatus was Stiles’ Trichromator at N.P.L. (STILES,1955). The main modification from Stiles’ conditions was the enlargement of the angular subtense of the circular field from 10’ to 1.5”.Otherwise it was similarly divided symmetrically horizontally, the test stimuli occupying the upper half. The circular surround, which was physically part of the lower field (Sruss, 195.5),was also enlarged proportionately to a diameter of 20”. For a few experiments with extrafoveal vision the subtense of the field was masked down to 5’ (circular, divided symmetrically horizontally, with a dark surround). A fixation spot 20’ in subtense appeared at an angle of either 7’ or 14’ from the centre of the field on the temporal side. (The right eye was used throughout.) This spot had the spectral composition and i&minance of the Iower half of the field. The instrumental stimuli were at 470.6 nm (B), and 558-2 nm (Y), as chosen by STILES(1957). Other detaiis such as the band-pass of the inept were exacdy as STUSS(1955) describes. The radiant fluxes were measured directly after each match with a calibrated photomultiplier, as described by CRAWFORD (1965). The linearity and stability of this type of detector are not above suspicion (CRA\KFORD, 1962). However, the present cell, which had been used in various tests of additivity in photometry (PALMER, 1966) and calorimetry (LOZANO and PALMER,1967), invariably proved linear to within + 2 per cent over a wide range of iltuminances.
Luminance Matching Functions
565
Measuring luminance matchingfimctions The following method was developed to find the particular mixture of yellow and blue stimuli which would match a given test stimulus in luminosity at all levels. The test stimulus was set at a high level and mixtures of yellow and blue in various proportions were matched to it in luminosity by direct comparison, with central fixation. The last proved unimportant (see later). The instrument controls were ganged together (Srnm, 1955) so that matches were easily made without altering the proportions in the mixture. After each match the setting was disturbed at random, and matches were repeated until three or four in good agreement were obtained in succession. The arithmetic means of the accepted readings were calculated. The proportion of blue stimulus required to match a unit amount of test stimulus was plotted against the corresponding quantity for yellow. The resultant graph was generally a curve, ranging from all-yellow, no-blue to all-blue, no-yellow, as in Figs. 1,4 and 5 which will be described later. This procedure was repeated at successively lower levels, with allowance for progressive dark-adaptation, to produce a family of curves with various slopes related to the Purkinje shift. According to the hypothesis, these curves should intersect at a unique point, which should indicate the co-ordinates (y,, b,) for equivalence, in terms of the experiment, with the test stimulus. For monochromatic test stimuli, these co-ordinates are the spectral stimulus values, the set of which constitutes the luminance matching functions. Curves could perhaps be fitted mathematically to the data. However, the theoretical form of the curves is as yet unknown (and indeed TESTIERand BLOTTIAU[1951] found several varieties), so the intersection point was estimated by eye from freehand drawing. The justification for this procedure will be evident when the graphs are discussed. The analogy with the colour matching functions suggests that a red or orange test stimulus will require “desaturation” with blue before the match can be made. In such a case, the yellow instrumental stimulus was set to the required level and matched successively with various mixtures of the test and the blue stimuli. Similarly, violet test stimuli required admixture with the yellow stimulus. Experiments The luminance matching functions were investigated for the author’s right eye, which had normal colour vision and normal photopic and scotopic sensitivities. Detailed measurements were made by the above method for stimuli at 530 nm, 440 nm and 620 mn, the last two requiring admixture, as just described. The numerous illuminance levels extended from 100 to O-01 trolands. Experiments were made at these three wavelengths with the 15” field viewed centrally, and with the 5” field viewed centrally and eccentrically at 7’ and 14’ on the nasal side, by using the small fixation spot already mentioned. The effects of adaptation to blue, green and red lights were cursorily investigated for the above three wavelengths in the 15” field. The adapting light was a flash torch covered with layers of diffusing paper and the appropriate coloured filter. The blue filter was Wratten 47B, with a maximum transmission at 440 nm. the green was Wratten 74, peaking at 540 mn, and the red, Wratten 70. transmitting above 650 nm. This source was held close to the eye for about 1 min. The illuminance was about 20 times that of the instrument field. After adaptation, the match was made as quickly as possible, with readaptation before the next match. With this variation, the procedure was as before. Spectral wavelengths from 380 nm to 740 nm at 30 run intervals were matched by the basic technique but
only in the 15”field, and sometimes to save time, at only two levels, namely 1 troland and @03trolands. The points of intersection of the resultant graphs were estimated by eye to provide the stimulus values for the luminance matching functions. The functions were also measured for some wavelengths in the 15” field, by an analogue of Maxwell’s method for colour matching (CRAWFORD, 1965). Mixtures of the instrumental stimuli in the test field were 6rst matched (in terms of the experiment) to a standard white (w) in the mixture field. This was N.P.L. Bluish White (Srr~ss. 1955). Then the soectral stimuli were simihuly matched to the standard by adding to each either the yellow or the blue stimulus as
appropriate. The equations are analogues of those described by CRAWFORD (1965):
m = Y’I (Y)
+ b’, 03)
whence
so that In practice either y, (I\) or b,(A) was zero, only the appropriate
stimulus being added to the test stimulus.
566
D. A. PALMER
Four non-mon~~omatic lights, which were those numbered 4,5,7,8 in a previous publication (Pw, 1967), were aiso matched by the technique in the 15”field by usingnumerous ili~~~~levels. The analogues to the tristirnuius values in cotorirnetry, which wiIl be termed stimulus values in this account, were found both by the direct method of observation and by the Maxwell-type of method, for comparison. To test additivity, they were also calculated as the integrals of the products of the luminance matching functions and the spectral power distributions. The last were measured by running a slit through the spectrum of the instrument whilst recording on the photomultiplier (PALMER,1966). If additivity holds, the mathematically predicted values should agree with the observed. RESULTS
The results appear in the figures and the table. Figure 1 shows the proportions of yellow and blue ins~ument~ stimuli in various mixtures which matched in luminosity a monochomatic stimulus at 530 nm in the 15” field, centrally viewed. The quantities (yr, fi,) are normalised for unit power of the test stimulus. Each graph was obtained at the ilfuminance indicated
PK.?.I. Mixtures CylbJ which matched a unit amount of the stimulus at 530 nm in the 15”fieId at various illuminances indicated at the feet of the respective graphs.
at its foot. It can be seen that the precision of observation is good; variations greater than f 10 per cent between matches were rarely found. Figure 2 is simiIar to Fig. 1 except that the monochromatic test stimulus is at 440 nm. The yellow stimulus was added to the test, and is shown conventionally as a negative quantity. Figure 3 is as Fig. 2, but for the test stimulus at 620 nm. The blue stimulus is now added to the test. Figure 4 is as Fig. 1 but for the 5” field at the peripheral angles 0”, 7’ and 14”, as indicated beside the respective curve. The points of inflection for the three conditions of observation are indicated by stars. These rest&s are for the 530 nm stimulus. Corresponding experiments
567
Luminance Matching Functions
C
-3
-I
-2
I
0
YI FIGS.2. As Fig. 1, but for the stimulus at 440 MI. Note the negative values for y,.
o-
-0.1
-
-0.2
-
bl
-0.3 l n ”
^C
V’3
I’0
_
YI FIG. 3. AS Fig. 1, but for the stimulus at 620 run. Note the negative values for b,.
D. A. PALMER
FIG. 4, As Fig, 1, but for the 5’ field at various peripheral angles which, together with the are indkated at the foot of each graph. The intersections for the three peripheral m&s are starred, and should be compared with that in Fig. 1.
illutiances,
0
I
2
3
4
5
6
7
3
9
IO
YI
FE. 5. As Fig, 1, but for mat&es after adaptation to blue, green and red IigMs @, g, rl. 3%~ interstctions for the three umditims of adaptation are star&, and &mid be cumparcd with that in Fig. 1.
569
Luminance Matching Functions
A.
nm
Fro. 6. Luminan Ce matching functions measured by the dixect method (solid points) aad the MaxweWpemethod (open points). n , yr(A); 0,6&l). The vertical lines indicate the instrumental stimuli. I.0 -
0.1
V 0.01
.
0.001
\
L
1 400
I 600
I 500
A.
p(:)
\ . \\ \ ‘r I
700
nm
tineai’ transforms of the huninance matching function compared with v’ (A) (---), and J’, (A)(- - - -) (lower curve). ., 096 1, (A) + 0.6 6, (A);& 0.7 g,(A) : 0-z 61 (A); n , O-7YI (A) + 0% (A). The last two are indistinguishable above 600 MI, where also VI0 (A) sc V, (A). VJt. IO/74
5-m
D, A. PA.WER
with the 440 nm and 620 nm stimuli, which are not iflusfrated here, displayed a &m&r slight variation of the point of intersection with field size and peripheral angle. Figure 5 illustrates the effect of adaptation to blue, green and red lights, for matches wit& 530 nm stimulus in the 15” field, centrally viewed, as for Fig. f . The intersections for the three sets of graphs are indicated by stars. Corresponding experiments with stimuli at 440 nm and 620 nm displayed a similar slight variation of the point af intersection with adaptation. Figure 6 shows the Iumiuance rn~~~~~~~functions measured in the f 5” field by the direct method and the Naxweif-type method. Figure 7 shows various linear transforms of these functions compared with some C.I.E. V’(X)and V(X)curves. The Table contains the stimulus values for the four non-rnonoch~~t~~ Iights. The values calculated from the “direct’” luminance matching functions and the spectrai power distributions are compared with those observed by the direct method and by the Maxwefttype method. DHGUSSION
The very distinct points of intersection of the graphs in Figs. i-3, which are only a selection from similar results for some 20 lights, show that the proposed stable luminosity matches may indeed be made, at least by one observer. For the three fights tested in detail, the matches were almost unaffected by field size. A very sfight, and probably insignificant variation in the point of intersection with per~pherai angle is shown in Fig. 4, which shotid be compared with Fig. 1, for the $5” field. These conclusions were borne out by the visual appearance of the fieid in the mesopic range. A partial central scotoma usually appeared in the side of the field with the greater scotopic luminance. The scotama disappeared when the field was matched both in scotopic and photopic hnninance, so that a ~~rn~~~s~~~ match was then ~tab~~~~ed ah along the dividing line. Such a match is obviously hardly affected by field size or peripheral angle, a least within the ranges examined here, and hence the fixation need not be carefully controlled. Adaptation had a larger, and probably more significant effect (Fig. S), which was not investigated sufEcient.ly for other tonelusions to be drawn. It is futile to attempt to measure l~~nan~e matching functions with a small field centrally viewed, because there is then no Purkinje shift. Nevertheless, the C.I.E. V,(h) curve for a 2” SeId can also be regarded as a transform of these functions, and thus part of the same scheme (Fig. 7). This coincidence is closely related to the apparent invariance of the Iuminance matching functions with respect to field size and peripheral angle. Another related manifestation is that V&X) happens to be a linear transform of Y’(X) and V, (X) fPkU.XInR,1868). The Maxwell-type of measurement produced results very sim.iIar to those of the direct method (Fig. 6) except that the Maxwell ji, (A) function is significantly the larger in the region between the stimuli. The general agreement between the methods suggests that the quantities in the matches may be manipulated ~~braica~y to predict other matches made with different stimuh. fn further support of this conch&on, hnear transforms of the tuminante functions can be found (Fig. 7) which cXosefyresemble Y’(h) and Y(h), That these pure rod and cone responses can be derived from mesopic data greatly supports the theory. The dire& tests of addit~~ty with no~~rno~~hrornati~ rights ffable f ) display s~~~~~nt differences between the stimulus values found by the three methods, that is by calculation
Luminance Matching Functions
571
from the “direct” functions, and by “direct” and Maxwell-type observations. The “Maxwell” and the blue stimulus values are particularly in disagreement. It should be noted that the largest percentage discrepancies occur in the smallest stimulus values e.g., in B for lights 7 and 8, and thus are not of great significance. Although the luminance matching functions are not additive within observational precision (generally about &lo%, as already mentioned), they are probably little worse in TABLE 1
Stimulusvalues
Test light
Observed
Calculated
4 (Green) 5 (white) 7 (Purple) 8 (Yellow)
Y 0.539 B 0.726 Y 0.491 B O-295 Y 0.170 BOG45 Y 0.610 BO-099
Observed discrepancies from calculation (“A
Direct
Maxwell
Direct
Maxwell
056 0.66 0.48 0.27 0.17 0.033 0.56 0.094
0.35 0.81 0.38 0.33 0.14 0.054 0.47 0.14
+4 -9 -2 -8 0 -27
-35 +12 -22 +11 -17 f20 -23 f49
this respect than their analogues, the large-field colour matching functions (LOZAKO and PALMER,1967, 1968), and the V(h) curve (PALMER,1968). On the other hand, the required match once established in given field conditions is unaffected, within the precision of observation, by changes of illuminance. The curvature of the photopic graphs (e.g. Fig. 1) is clearly the consequence of photometric non-additivity. Nevertheless, the curves all intersect precisely at the point defined by their more linear scotopic counterparts. This result obtains even with large colour differences. Such observations suggest a general theorem. Three or more stimuli may be arranged to produce two spectrally different lights whose luminosities will match by direct comparison, within the precision of observation, at all levels. For three stimuli the match is unique. (Neutral attenuation is assumed, and field size, fixation and adaptation should be controlled as they may have slight effects.) A generalisation drawn by induction from one person’s observations of a limited range of test lights must be tentative. However, a corollary has been widely tested. In previous experiments (PALMER,19661968), 19 individuals had matched in luminosity several nonmonochromatic stimuli to the range of spectral stimuli at various illuminances. Each non-monochromatic stimulus could always be matched within observational precision by a particular monochromatic stimulus in a characteristic ratio, whatever the level. Experiment thus verifies a special case of the theorem, whose conditions are fully met, since any nonmonochromatic stimulus is composed of at least two stimuli. (The proportions of the components cannot be altered without changing the stimulus, so the stable match can be made only with a particular spectral stimulus.) The near-additivity of this type of match and its partial invariance with respect to adaptation obviously suggest Grassmann’s Laws in calorimetry (WRIGHT, 1969), which could perhaps be reframed to include such stable luminosity matches. Since V(X)can serve
572
D.A.
PALMER
as one of the colour matching functions, perhaps only V’(h) need be added to the colour matching functions to produce a set of four functions which, for the normal observer, might be necessary and sufficient to define matches stable in colour and luminosity over a wide range of illuminances. The suggestion remains speculative until tested by experiment. The visual receptors might change their responses to colour, whilst maintaining their response to luminance. The suggested matches might then break down in colour at various levels. Such discrepancies have been found in corresponding experiments with a deuteranope (PALMER,1969). To conclude with considerations for practical photometry, it appears unnecessary to derive the fundamental curves from measurements in pure scotopic and photopic conditions. Luminance matching functions can be obtained from mesopic measurements, with little attention to field size and peripheral angle. Standard photocells need not be calibrated exactly with respect to defined v’(h) and Vo\) curves. Any linear transforms of the luminance matching functions would serve in practice, except those with negative lobes. This freedom might facilitate the design of spectral correction filters (CRAWFORD,1962). To complete this scheme, the luminance of any test light at any level, under given field conditions, would have to be known as a function of the integrals of the luminance matching functions with respect to the spectral power distribution. A method of deriving this function experimentally has already been outlined (PALMER,1967), and as it does not necessarily require monochromatic test lights, it differs fundamentally from the apparently similar process of determining the so-called mean mesopic V(A)functions (HOUGHand RUDDOCK, 1969). The effect of field size can be allowed for by a parameter in an empirical expression for the luminance (PALMER,1968). The disadvantage of this universal system is that two physical measurements would generally have to be made to determine each luminance, though practice might teach when one would suffice. The luminance matching functions of the average Observer, which are not yet available for discussion, might therefore be applied to practical photometry. The functions of 15 individuals have since been determined, with the intention of publishing them shortly. Acknowfe&emenfs-Dr. B. H. CRAWFORD’Sadvice is gratefully acknowledged. The work described in this paper formed part of the officialresearch programmeof the National Physical Laboratory.
REFERENCES CRAWFORD, B. H. (1962). Physical photometry. N.P.L. Noteson Applied Science No. 29. H.M.S.O., London. CRAWFORD, B. H. (1965). Colour matching and adaptation.
Vision Res. 5, 71-78. HOUGH, E. A. and Rt.mooorr, K. H. (1969). The parafoveal visual response of a tritanope and an interpretation of the Vh sensitivity functions of mesopic vision. Vision Res. 9, 935-946. LQZANO, R. D. and PALMER,D, A. (1967). The additivity of large-field colour matching functions. Vision Res. 7,929-937. L.OZANO,R. D. and PALMER,D. A. (1968). Large-field color matching and adaptation. J. opt. Sot. Am. 58, 1653-1656. PALMER,D. A. (1966). A system of mesopic photometry. Nature, Land. 209,276281. PALMER, D. A. (1967). The definition of a standard observer for mesopic photometry. Vision Res. 7,619-628. PALMER, D. A. (1968). Standard observer for large-field photometry at any level. J. opt. Sot. Am. 58,12961299. PALMER, D. A. (1969). Colour matches which include equality of scotopic luminance. Proc. 1st. Congress A.Z.C., Stockholm, 1969. Musterschmidt VerIag, Gottingen. In press. STILES,W. S. (1955). 18th Thomas Young Oration: the basic data of coIour-matching. Yearbook the Physical Society, pp. 44-65. STILES,W. S. (1958). The average colour-matching functions for a large matching field. Proc. 8th N.P.L. Symp. 1957,~~. 211-247,H.M.S.O.,London.
of
Luminance TESER, M. and B-u, antes photopiques.
F. (1951). Variations
Matching
Functions
des characteristiques
573
photometriques
de l’oeil aux lumin-
Rev. d’Opt.. Paris, 30, 309-322. Wm~trr, W. D. (1969). The Meusurement of Colour, Hilger & Watts, London Abstract-Three or more different stimuli may be arranged in a photometer field to produce two spectrally different lights which match in luminosity at all levels. A monochromatic stimulus may be so matched by a characteristic mixture of two instrumental stimuli, the quantities of which for a unit test stimulus may be defined as the spectral stimulus values. The set of these values constitutes the luminance matching functions, which may be manipulated algebraically in order to predict the approximate stimulus values of any test light of known constitution. Y’(h) and Y(h) may be derived as linear transforms of the functions, from mesopic data. Applications to practical photometry are suggested. R&amGOII peut combiner dans un champ photometrique trois stimuli ou plus pour produire deux lumi&res spectralement diff&rentes qui sont de meme luminosite a tous les niveaux. Un stimulus monochromatique peut Metreainsi &alid avec un melange caractiristique de deux stimuli instrumentaux, dont les quantit&s pour un test stimulus unite peuvent ftre deli&s comme la valeurs de stimulus spectrales. L’ensemble de ces valeurs constitue les fonctions d&alit& de luminance, qui peuvent 6tre manipultes algebriquement pour predire les valeurs approchtcj de stimulus pour toute lumiere test de composition connue. v’(X) et Y(h) peuvent etre obtenus par transformations lin&ires de ces fonctions a partir de don&s m&.opiques. On suggere des applications B la photometric pratique. Zusammcnfassurrg-Drei oder mehr als drei verschiedene Reize konnen so in einem photometrischen Felde dargeboten werden, dass zwei spektral verschiedene Lichter, deren Helligkeiten einander an allen Niveauxgleich sind, erhalten werden konnen. Ein einf&rbiger Reiz kann somit einer kennzeichnenden M&hung zweier Instrumentreize, deren G&se auf Grund einer Reizeinheit als Spektralreiz ausgedrtickt, ist, gleichgesetzt werden. Die Wertgruppe gleicht den Helligkeitsfunktionen, welche algebra&h so gehandhabt werden kiinnen, das der ungef&re Reizwert aller Testlichter, deren Z usammensetzung bekannt ist, vorausgesagt werden kann. Y (I\) und V(A) k6tmen als Lineartransforme dieser Funktionen von mesopischen Ergebnissen abgeleitet werden, Es werden Vorschlige fur die Ausntitzung der Methode in der praktischen Lichtmessung gemacht. Pearome - Tpa mm 6onee p asnmmbtx crn~ym MOQT 6~13 TSLKLI~HMB none &ITOM~T~~, ¶TO IIpOjQIIKpyioT CIIeKTpaJIbIIO paSIE¶IbIe CBeTOBbIe pa3IIpaXtETeJIE, KOTOpble M0I”y-r ~paarni~aT%c~ no ~~KOCTII Ha BceX YpOBIISIX. MOHOXpOMaTIiKeCKIItJCTI~.&+TMOX~T Bt.t~b. TaKEM o6pa3OM, y-paBIieH CO CMeCbIO JIByx EXCrpyMeIITaJIbIIbIX CTIIMyJIOB, ROJIH¶ecrBa K0~0pbIX ALIBIemor TecTOBOrO CTIIMyna MOI-yT 6brrb OIIpeneJIeIKbf KaIC BCrIeIcTpaJIbaoro CTIIM~M. Ha6op T~KIIX B~IRSAII 06pa3yer &rmmrn cpaanemfa nprocrn, KOTOpbtMkI MOXSIO OuepHpoBaTb a~n-e6pamtecrcn, C IIeJIbIO I.IpeACKa3aTb ITpII6JIE3IITelTbII&Ie B crn~yna nr~a mo6oro Tecroaoro cBeTa n3Becrnoro cocra~a. v’ (A) II V(A) ~oryr
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