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Which two lights that match for cones show the greatest ratio for rods?
Vi&n
Rrr.
Vol.
16, pp. 303-307.
Pergamon
Press
1976.
Printed
in Great
Edain.
WHICH TWO LIGHTS THAT MATCH FOR CONES SHOW THE GREATEST RATIO FOR RODS? R. W. RODIECK Brain Research Unit, Department of Physiology, University of Sydney, Sydney, Australia (Receiued 24 February 1975: in revised form 12 May 1975) Abstract-Within the 400-700 nm spectral range, the light having the maximal relative scotopic value is formed by mixiig a 504- with a 700-nm tight. The light with the minimal relative scotopic value is formed by mixing a 400- with a 629-nm light. When these two mixtures are matched for cones they differ for rods by a ratio of 14.0.
Two spectrally different lights, though physically distinct, may nevertheless appear indistinguishable. In cone vision, this occurs when each of the three cone types is equally stimulated by the two lights. Outside the fovea, where both cones and rods are present, these lights will be seen by cones alone either during the early phase of dark adaptation or above rod saturation. For all but the reddest lights one can place a neutral filter before them so as to reduce their intensities below the thresholds of the cones, but still above the dark-adapted rod threshold. Then it is only with rods that we see these spectrally distinct lights. Though matching for each of the cones, they will in general differ for the rods, one appearing brighter than the other. A neutral filter can be placed before the brighter so that the two lights will equally stimulate the rods and they will again appear indistinguishable. The density of this neutral filter will depend upon the two lights chosen. What is the greatest density ever required and what two lights give it? Any solution to this problem requires some specification of the conditions that allow two lights to match for cone vision and for rod vision. The spectral sensitivities of the three cone types and of the rods would suffice. These curves are not known for the cones, but linear combinations of them are; these are the 1964 CIE color-matching functions a,,(A), j,dA) and Z,,,(A), which govern extra-fovea1 color matches (Wyszecki ‘and Stiles, 1967, p. 274). Using these functions, one can test if two lights will match by means of the following procedure. For a light of spectral energy distribution E(A). calculate the following quantities: X=
Y= Z =
I I
I
E(A)B,o(A) dh
may be calculated by means of another function, related to rod spectral sensitivity. and termed the relative scotopic luminous eficiency function, V’(A): V’ =
I
Two lights, with spectral energy distributions E,,(A) and E*(A) respectively, will match for all three cones when: x,=xb
Yo= Yb
z. = Zb where X. =
but
I
in general
E,(A)~,~(A)~A,
they
dh.
These three quantities, termed tristimulus values, serve to characterize the effect of the light on the three cones in the same way that the photon catches of the cones could potentially be so used. The relative magnitude of the rod stimulation, V’, 303
match
for rods, intensity
ratio:
The problem is now essentially a mathematical functions E.(A) one: “what two positive-valued and E*(A), which have identical tristimulus values, maximize d?” The solution presented here makes use of a generalization of the CIE chromaticicy coordinates x and y. These quantities are obtained from the tristimulus values in the following way: X
E(A)&(A)
will not
etc.
Vh # Vb, and one can define a scotopic
x=x+Y+z’
E(A)j%i(A) dA
E(A)V’(A)dA.
Y Y=X+Y+Z-
For any light these values define a point on a chromaticity diagram as in Fig. l(a). This diagram shows the locus of monochromatic lights. as well as the points corresponding to two arbitrary lights E,,(A) and EodA). The tristimulus values X,. Y and Z are each proportional to the intensity of the light: but the chromaticity coordinates, being ratios of tris-
301
R.
“I
“1
Cc)
h’.
Cd)
Fig. 1. Addition of tights in CIE (x. y) space [a, b], and in the (x. y. u) space described in the text [c, d]. timulus values, are independent of intensity. Conversely, two lights that map to the same point on the diagram can be made to match by varying the intensity of one of them. When two lights are mixed, their sum maps to a point that lies on a straight line drawn between their corresponding points, as shown in Fig. l(b). It is convenient to here define a third axis, termed the scotopic coordinate, that plots the quantity: V’
(2)
u=x+Y+z
for then the sum of two lights also maps to a point that lies on a straight line drawn between their corresponding points in (x, y, u) space. Here is the proot: Let: E.(A) = E,,(A)+
Roo~icti
Sow if the quanrirv (X2, - I-‘, - Z,:! IZ idnjldered the “mass” of I;ght E, IA ). etc.. then these equations are seen. for each coordinate, to give the center of mass of the sum of lights E,,i,\) and Lfh ). They thus define the center of mass in (.u. .i‘, v) space as well, which, of course, lies on the straight line between the points in space representing E,,(h) and E,?(A) (Fig. l(c)). The argument easily extends to the sum of n lights; the resultant lies at their center of mass in (x. )‘. L.) space. The locus of monochromatic lights is plotted in (x, y. t.) space in Fig. ‘. _, this curved line is of particular importance to the question considered here. This is because, in that all lights can be represented as a limiting sum of monochromatic lights, a point in (1. :. c ) space can represent a real light only if it can be the center of mass of a series of masses placed along this spectral locus. Why is (x. y, c) space useful to our question? Consider again two lights, E,(A) and G(A), that match for cones. They will necessarily have the same x and y values but will in general differ for their c’ values; in x, y, L’space one will lie above the other as shown in Fig. l(d). Now from equations (1) and (2): d
=s= Vb
ti,(X. + YG+Za) L’hfXh + Yh fZh)
but when the lights match for cones they have identical tristimulus values, so that: (X,+Y,tZ,)=(X*tYb+Zh) and therefore: d=$.
E&)
then: Vb- _/ E,(h)V’(A)dA
Urn =
x,
+
= j- [E,,(h)+E,?(A)lV’(A)dh
= v:l+
V.lZr etc.
Vh, + Vi2 Vb Y. + 2, = (X,, + Y,, + Z,,) + (X02 + Ya?-!--L)
=(x.I+~:+z.,)x
[
(x.,+
(X.1+ Y.I+zat) Y,,+Z.,)+(Xa?+ Y.?+L)
(X.2 + Y,? + Z.2) VA Y‘C+-m + (X02 + Y,z + 202) x C(X*, + Y,, +Z,,)+(X,:+ ~.=~‘.,.k+u.z(l-k)
where
likewise x. = X.21. k+x.z(l-k) and ye = y,t . k + y,z(l - k).
I
I
Which two lights that match for cones show the greatest ratio for rods?
Fig. 2. The spectral locus in (x. y, c) space.
Thus we seek two lights that have the same chromaticity coordinates but differ by the greatest ratio for their scotopic coordinates. The domain of all real lights in (x. y, u) space is a solid on whose surface lies the spectral locus. We need to consider the upper surface of this solid, which, for each chromaticity point, gives the maximal scotopic value, and the lower surface, which is the locus of the minimal scoptopic values. Here is a method for determining the compositions of the lights that give these maximal or minimal values. For a given chromaticity point construct a plane tangent to the maximal surface. No point on the spectral locus can lie above this plane, for if it did then a larger scotopic value could be found and the surface would not be maximal. On the other hand the spectral locus cannot lie entirely below the plane, for if this were true then no combination of masses below the plane could give a center of mass lying on the plane. Thus the spectral locus must touch the plane at one or more points. A limiting case occurs when it touches at but one point and corresponds to the condition where the chosen chromaticity point lies on the spectral locus at that point. For chromaticity points within the spectral locus, the plane must touch the spectral locus at two or more points. If it touches at but two points then the maximal scotopic value is realized by a mixture of the corresponding pair of monochromatic lights. Other chromaticity points along the line drawn between these two spectral points have maximal scotopic values that can be realized by other mixtures of the two monochromatic lights. If the plane touches the spectral locus at three points, then, at all chromaticity points within the triangle defined by the three points, the maximal scotopic \ :II~ICic realized by a mixture of the three
303
corresponding monochromatic lights. In general the plane will not touch the spectral locus at more than three points, since all such points would have to happen to be coplanar, but were this to happen then any three spectral points that include a given chromaticity point can be used to realize the maximal scotopic value at that point. Similar reasoning applies to the realization of the minimal surface. Thus, at a point within the spectral locus, a maximal or a minimal scotopic value can always be realized by either two, or. at the most, three monochromatic lights. Conversely, the maximal or minimal surfaces can be found by pressing a plane against the upper or lower surface of the spectral locus and noting where they touch. I constructed a metal model of the spectral locus from 400 to 700 nm and placed a flat surface against it. When the surface was brought down from above and pushed against the model in different ways it was always found to touch at but two points, one of which was always at the long-wavelength limit of the spectral locus. Thus the maximal scotopic value at any given chromaticity point can be realized by a mixture of 700-nm light with another monochromatic light. When the tlat surface was pressed onto the spectral locus from below it was always found to touch at but two points, one of which was always at the short-wavelength limit of the spectral locus. Thus the minimal scotopic value at any given chromaticity point can be realized by a mixture of 400-nm light with another monochromatic light. The ratio of maximal to minimum scotopic values can thus be calculated for each point on the chromaticity diagram. The logarithm of this ratio may be used to express this difference in density units and isograms of constant density are shown in Fig. 3. The largest ratio of all shows a density difference of 1.15
x
Fig. 3. CIE (1964) (x. y) chromaticity diagram showing maximal scotopic intensity ratios for lights that match for cones. The number next to each isogram corresponds to the logarithm,, of this ratio and thus gives the equivalent density value. The insert at the upper right shows the method for realizing the two lights at any chromaticity point.
R. W.
306
RODIECI~
Id = 14.0) and occurs when cones match a mixture of 700- and SO-l-nm lights with a mixture of 400- and 619-nm lights. IYTESSITY
RELATIOSS
The lights having the largest scotopic ratio are composed mainly of long-wavelength photons, to which the rods are relatively insensitive. It is therefore necessary to check that the scotopically brighter of the two can lie below cone threshold. Likewise it is necessary to determine how intense the dimmer of the two would have to be to saturate the rods. and to compare this intensity with that at which color matches are altered by pigment bleaching. These questions can be answered by comparing the scotopic and photopic luminances of these stimuli. The scotopic luminance, L ‘, is defined as:
L'= KLV' = KA,
i
E(A)V’(A)dA X
where K,i, is a constant (1745 scotopic Im W-‘) (Wyszecki and Stiles. 1967, p. 384). Let LA, be the maximum scotopic luminance relative to the photopit luminance at a given chromaticity point, and Lki. be the minimum scotopic luminance relative to the photopic luminance at the same point. When these two lights are matched photopically the logarithm of their ratio:
Fig. 4. CIE (1964) (x, y) chromaticity diagram showing isograms of the relation D,, = 10g,~(L &IL *I where L * is the reference photopic luminance at a given chrornatlcity point and LL, is the corresponding maximal scotopic luminance at that point.
D = log,+-= Ill,”
is
the scotopic intensity ratio plotted in Fig. 3. There is no standard way of calculating the extrafoveal photopic luminance, L*. Stiles and Wyszecki (1973) have used the following definition:
J
L*=Kz
-
E(h)Ym(A)d
where K 2 is a constant (680 photopic 1m WY’). The logarithm of the ratio of the maximal scotopic luminance to the photopic luminance: D rrai
=
log,0
s
is plotted in Fig. 4. while Fig. 5 plots: D,i. = logo e. The calculated values in Figs. 3-5 are not mutually independent, for at any point it follows from their definition that: D = D,,
-
D,i,.
These figures give the maximum and minimm scotopic values at any chromaticity and thus provide a general solution to this question, which has been investigated in a specific way by Stiks and Wyszecki (1973).
Fig. 5. CIE (1964) (x, y) chromaticity diagram showing isograms of the relation iJ,+” = log. (L A./L *) where L * is the reference pbotopic luminance at a given ehromaticity point and L&. is the comtspmding minimal scotopic luminance at that point.
ROD AND CONE THRBEOLDS
For a given light there is no fixed relation between rod and cone thresholds: this difterence depends upon the size, duration and locationof the test spot. For a 507~nm spot of relatively large size and long duration the rod threshold lies about 3 log units below the cone threshold (e.g. Wyszecki and Stiles,
Which two lights that match for cones show the greatest ratio for rods’?
1%7, p. 569). Under the same conditions, but with the scotopically brighter light substituted for the 507~nm one, would the rod threshold still lie below the cone threshold? From Fig. 4, at the point giving the maximal scotopic ratio D,, = -0.23, whereas for a 507-nm light D, = +066. Thus, under otherwise identical conditions, substitution of the scotopically brighter light for the 507-nm light would raise the rod threshold by some 0.9 log units relative to the cone threshold. The rod threshold would thus still lie below cone threshold, by about 2 log units.
307
bleach away most of the visual pigment and would alter the color match (Wystecki and Stiles, 1967, p. 125). Therefore, in practice. the match for cones would have to be established during the early phase of dark adaptation to a bleaching exposure, when the cones had recovered their sensitivity, but before the rods had recovered theirs. Acknowledgements-1 would like to thank W. A. H. Rushton for his useful comments on a draft of the manuscript.
ROD SATURATION
Rod saturation occurs at a background level of about 3.5 x 10’ scotopic td (Aguilar and Stiles, 1954). From Fig. 5, at the chromaticity point having the largest scotopic ratio, the scotopically dimmer light lies 1.38 log units below the corresponding photopic value. The scotopically dimmer light would thus have to have a photopic intensity of 8.3 x 10’ photopit td. A steady background of this intensity would
REFERENCES
Aguilar M. and Stiles W. S. (1954) Saturation of the rod mechanism of the retina at high levels of stimulation. Optica Acta 1. 59-65. Stiles W. S. and Wyszecki G. (1973) Rod intrusion in large-field color matching. Acta chromatica 2.155-163. Wyszecki G. and Stiles W. S. (1967) Color Science. Concepts and Methods, Quantirariue Dara and Formulas. Wiley, New York.
Rrr.
Vol.
16, pp. 303-307.
Pergamon
Press
1976.
Printed
in Great
Edain.
WHICH TWO LIGHTS THAT MATCH FOR CONES SHOW THE GREATEST RATIO FOR RODS? R. W. RODIECK Brain Research Unit, Department of Physiology, University of Sydney, Sydney, Australia (Receiued 24 February 1975: in revised form 12 May 1975) Abstract-Within the 400-700 nm spectral range, the light having the maximal relative scotopic value is formed by mixiig a 504- with a 700-nm tight. The light with the minimal relative scotopic value is formed by mixing a 400- with a 629-nm light. When these two mixtures are matched for cones they differ for rods by a ratio of 14.0.
Two spectrally different lights, though physically distinct, may nevertheless appear indistinguishable. In cone vision, this occurs when each of the three cone types is equally stimulated by the two lights. Outside the fovea, where both cones and rods are present, these lights will be seen by cones alone either during the early phase of dark adaptation or above rod saturation. For all but the reddest lights one can place a neutral filter before them so as to reduce their intensities below the thresholds of the cones, but still above the dark-adapted rod threshold. Then it is only with rods that we see these spectrally distinct lights. Though matching for each of the cones, they will in general differ for the rods, one appearing brighter than the other. A neutral filter can be placed before the brighter so that the two lights will equally stimulate the rods and they will again appear indistinguishable. The density of this neutral filter will depend upon the two lights chosen. What is the greatest density ever required and what two lights give it? Any solution to this problem requires some specification of the conditions that allow two lights to match for cone vision and for rod vision. The spectral sensitivities of the three cone types and of the rods would suffice. These curves are not known for the cones, but linear combinations of them are; these are the 1964 CIE color-matching functions a,,(A), j,dA) and Z,,,(A), which govern extra-fovea1 color matches (Wyszecki ‘and Stiles, 1967, p. 274). Using these functions, one can test if two lights will match by means of the following procedure. For a light of spectral energy distribution E(A). calculate the following quantities: X=
Y= Z =
I I
I
E(A)B,o(A) dh
may be calculated by means of another function, related to rod spectral sensitivity. and termed the relative scotopic luminous eficiency function, V’(A): V’ =
I
Two lights, with spectral energy distributions E,,(A) and E*(A) respectively, will match for all three cones when: x,=xb
Yo= Yb
z. = Zb where X. =
but
I
in general
E,(A)~,~(A)~A,
they
dh.
These three quantities, termed tristimulus values, serve to characterize the effect of the light on the three cones in the same way that the photon catches of the cones could potentially be so used. The relative magnitude of the rod stimulation, V’, 303
match
for rods, intensity
ratio:
The problem is now essentially a mathematical functions E.(A) one: “what two positive-valued and E*(A), which have identical tristimulus values, maximize d?” The solution presented here makes use of a generalization of the CIE chromaticicy coordinates x and y. These quantities are obtained from the tristimulus values in the following way: X
E(A)&(A)
will not
etc.
Vh # Vb, and one can define a scotopic
x=x+Y+z’
E(A)j%i(A) dA
E(A)V’(A)dA.
Y Y=X+Y+Z-
For any light these values define a point on a chromaticity diagram as in Fig. l(a). This diagram shows the locus of monochromatic lights. as well as the points corresponding to two arbitrary lights E,,(A) and EodA). The tristimulus values X,. Y and Z are each proportional to the intensity of the light: but the chromaticity coordinates, being ratios of tris-
301
R.
“I
“1
Cc)
h’.
Cd)
Fig. 1. Addition of tights in CIE (x. y) space [a, b], and in the (x. y. u) space described in the text [c, d]. timulus values, are independent of intensity. Conversely, two lights that map to the same point on the diagram can be made to match by varying the intensity of one of them. When two lights are mixed, their sum maps to a point that lies on a straight line drawn between their corresponding points, as shown in Fig. l(b). It is convenient to here define a third axis, termed the scotopic coordinate, that plots the quantity: V’
(2)
u=x+Y+z
for then the sum of two lights also maps to a point that lies on a straight line drawn between their corresponding points in (x, y, u) space. Here is the proot: Let: E.(A) = E,,(A)+
Roo~icti
Sow if the quanrirv (X2, - I-‘, - Z,:! IZ idnjldered the “mass” of I;ght E, IA ). etc.. then these equations are seen. for each coordinate, to give the center of mass of the sum of lights E,,i,\) and Lfh ). They thus define the center of mass in (.u. .i‘, v) space as well, which, of course, lies on the straight line between the points in space representing E,,(h) and E,?(A) (Fig. l(c)). The argument easily extends to the sum of n lights; the resultant lies at their center of mass in (x. )‘. L.) space. The locus of monochromatic lights is plotted in (x, y. t.) space in Fig. ‘. _, this curved line is of particular importance to the question considered here. This is because, in that all lights can be represented as a limiting sum of monochromatic lights, a point in (1. :. c ) space can represent a real light only if it can be the center of mass of a series of masses placed along this spectral locus. Why is (x. y, c) space useful to our question? Consider again two lights, E,(A) and G(A), that match for cones. They will necessarily have the same x and y values but will in general differ for their c’ values; in x, y, L’space one will lie above the other as shown in Fig. l(d). Now from equations (1) and (2): d
=s= Vb
ti,(X. + YG+Za) L’hfXh + Yh fZh)
but when the lights match for cones they have identical tristimulus values, so that: (X,+Y,tZ,)=(X*tYb+Zh) and therefore: d=$.
E&)
then: Vb- _/ E,(h)V’(A)dA
Urn =
x,
+
= j- [E,,(h)+E,?(A)lV’(A)dh
= v:l+
V.lZr etc.
Vh, + Vi2 Vb Y. + 2, = (X,, + Y,, + Z,,) + (X02 + Ya?-!--L)
=(x.I+~:+z.,)x
[
(x.,+
(X.1+ Y.I+zat) Y,,+Z.,)+(Xa?+ Y.?+L)
(X.2 + Y,? + Z.2) VA Y‘C+-m + (X02 + Y,z + 202) x C(X*, + Y,, +Z,,)+(X,:+ ~.=~‘.,.k+u.z(l-k)
where
likewise x. = X.21. k+x.z(l-k) and ye = y,t . k + y,z(l - k).
I
I
Which two lights that match for cones show the greatest ratio for rods?
Fig. 2. The spectral locus in (x. y, c) space.
Thus we seek two lights that have the same chromaticity coordinates but differ by the greatest ratio for their scotopic coordinates. The domain of all real lights in (x. y, u) space is a solid on whose surface lies the spectral locus. We need to consider the upper surface of this solid, which, for each chromaticity point, gives the maximal scotopic value, and the lower surface, which is the locus of the minimal scoptopic values. Here is a method for determining the compositions of the lights that give these maximal or minimal values. For a given chromaticity point construct a plane tangent to the maximal surface. No point on the spectral locus can lie above this plane, for if it did then a larger scotopic value could be found and the surface would not be maximal. On the other hand the spectral locus cannot lie entirely below the plane, for if this were true then no combination of masses below the plane could give a center of mass lying on the plane. Thus the spectral locus must touch the plane at one or more points. A limiting case occurs when it touches at but one point and corresponds to the condition where the chosen chromaticity point lies on the spectral locus at that point. For chromaticity points within the spectral locus, the plane must touch the spectral locus at two or more points. If it touches at but two points then the maximal scotopic value is realized by a mixture of the corresponding pair of monochromatic lights. Other chromaticity points along the line drawn between these two spectral points have maximal scotopic values that can be realized by other mixtures of the two monochromatic lights. If the plane touches the spectral locus at three points, then, at all chromaticity points within the triangle defined by the three points, the maximal scotopic \ :II~ICic realized by a mixture of the three
303
corresponding monochromatic lights. In general the plane will not touch the spectral locus at more than three points, since all such points would have to happen to be coplanar, but were this to happen then any three spectral points that include a given chromaticity point can be used to realize the maximal scotopic value at that point. Similar reasoning applies to the realization of the minimal surface. Thus, at a point within the spectral locus, a maximal or a minimal scotopic value can always be realized by either two, or. at the most, three monochromatic lights. Conversely, the maximal or minimal surfaces can be found by pressing a plane against the upper or lower surface of the spectral locus and noting where they touch. I constructed a metal model of the spectral locus from 400 to 700 nm and placed a flat surface against it. When the surface was brought down from above and pushed against the model in different ways it was always found to touch at but two points, one of which was always at the long-wavelength limit of the spectral locus. Thus the maximal scotopic value at any given chromaticity point can be realized by a mixture of 700-nm light with another monochromatic light. When the tlat surface was pressed onto the spectral locus from below it was always found to touch at but two points, one of which was always at the short-wavelength limit of the spectral locus. Thus the minimal scotopic value at any given chromaticity point can be realized by a mixture of 400-nm light with another monochromatic light. The ratio of maximal to minimum scotopic values can thus be calculated for each point on the chromaticity diagram. The logarithm of this ratio may be used to express this difference in density units and isograms of constant density are shown in Fig. 3. The largest ratio of all shows a density difference of 1.15
x
Fig. 3. CIE (1964) (x. y) chromaticity diagram showing maximal scotopic intensity ratios for lights that match for cones. The number next to each isogram corresponds to the logarithm,, of this ratio and thus gives the equivalent density value. The insert at the upper right shows the method for realizing the two lights at any chromaticity point.
R. W.
306
RODIECI~
Id = 14.0) and occurs when cones match a mixture of 700- and SO-l-nm lights with a mixture of 400- and 619-nm lights. IYTESSITY
RELATIOSS
The lights having the largest scotopic ratio are composed mainly of long-wavelength photons, to which the rods are relatively insensitive. It is therefore necessary to check that the scotopically brighter of the two can lie below cone threshold. Likewise it is necessary to determine how intense the dimmer of the two would have to be to saturate the rods. and to compare this intensity with that at which color matches are altered by pigment bleaching. These questions can be answered by comparing the scotopic and photopic luminances of these stimuli. The scotopic luminance, L ‘, is defined as:
L'= KLV' = KA,
i
E(A)V’(A)dA X
where K,i, is a constant (1745 scotopic Im W-‘) (Wyszecki and Stiles. 1967, p. 384). Let LA, be the maximum scotopic luminance relative to the photopit luminance at a given chromaticity point, and Lki. be the minimum scotopic luminance relative to the photopic luminance at the same point. When these two lights are matched photopically the logarithm of their ratio:
Fig. 4. CIE (1964) (x, y) chromaticity diagram showing isograms of the relation D,, = 10g,~(L &IL *I where L * is the reference photopic luminance at a given chrornatlcity point and LL, is the corresponding maximal scotopic luminance at that point.
D = log,+-= Ill,”
is
the scotopic intensity ratio plotted in Fig. 3. There is no standard way of calculating the extrafoveal photopic luminance, L*. Stiles and Wyszecki (1973) have used the following definition:
J
L*=Kz
-
E(h)Ym(A)d
where K 2 is a constant (680 photopic 1m WY’). The logarithm of the ratio of the maximal scotopic luminance to the photopic luminance: D rrai
=
log,0
s
is plotted in Fig. 4. while Fig. 5 plots: D,i. = logo e. The calculated values in Figs. 3-5 are not mutually independent, for at any point it follows from their definition that: D = D,,
-
D,i,.
These figures give the maximum and minimm scotopic values at any chromaticity and thus provide a general solution to this question, which has been investigated in a specific way by Stiks and Wyszecki (1973).
Fig. 5. CIE (1964) (x, y) chromaticity diagram showing isograms of the relation iJ,+” = log. (L A./L *) where L * is the reference pbotopic luminance at a given ehromaticity point and L&. is the comtspmding minimal scotopic luminance at that point.
ROD AND CONE THRBEOLDS
For a given light there is no fixed relation between rod and cone thresholds: this difterence depends upon the size, duration and locationof the test spot. For a 507~nm spot of relatively large size and long duration the rod threshold lies about 3 log units below the cone threshold (e.g. Wyszecki and Stiles,
Which two lights that match for cones show the greatest ratio for rods’?
1%7, p. 569). Under the same conditions, but with the scotopically brighter light substituted for the 507~nm one, would the rod threshold still lie below the cone threshold? From Fig. 4, at the point giving the maximal scotopic ratio D,, = -0.23, whereas for a 507-nm light D, = +066. Thus, under otherwise identical conditions, substitution of the scotopically brighter light for the 507-nm light would raise the rod threshold by some 0.9 log units relative to the cone threshold. The rod threshold would thus still lie below cone threshold, by about 2 log units.
307
bleach away most of the visual pigment and would alter the color match (Wystecki and Stiles, 1967, p. 125). Therefore, in practice. the match for cones would have to be established during the early phase of dark adaptation to a bleaching exposure, when the cones had recovered their sensitivity, but before the rods had recovered theirs. Acknowledgements-1 would like to thank W. A. H. Rushton for his useful comments on a draft of the manuscript.
ROD SATURATION
Rod saturation occurs at a background level of about 3.5 x 10’ scotopic td (Aguilar and Stiles, 1954). From Fig. 5, at the chromaticity point having the largest scotopic ratio, the scotopically dimmer light lies 1.38 log units below the corresponding photopic value. The scotopically dimmer light would thus have to have a photopic intensity of 8.3 x 10’ photopit td. A steady background of this intensity would
REFERENCES
Aguilar M. and Stiles W. S. (1954) Saturation of the rod mechanism of the retina at high levels of stimulation. Optica Acta 1. 59-65. Stiles W. S. and Wyszecki G. (1973) Rod intrusion in large-field color matching. Acta chromatica 2.155-163. Wyszecki G. and Stiles W. S. (1967) Color Science. Concepts and Methods, Quantirariue Dara and Formulas. Wiley, New York.