We measured the relative contribution of rods and cones to luminance across a range of photopic, mesopic, and scotopic adaptation levels and at various retinal eccentricities. We isolated the luminance channel by setting motion-based luminance nulls (minimum motion photometry) using annular stimuli. Luminance nulls between differently colored stimuli require equality in a weighted sum of rod and cone excitations. The relative cone weight increases smoothly from the scotopic range, where rods dominate, to photopic levels, where rod influence becomes negligible. The change from rod to cone vision does not occur uniformly over the visual field. The more peripheral the stimulus location, the higher is the light level required for cones to participate strongly. The relative cone contribution can be described by a sigmoid function of intensity, with two parameters that each depend on the eccentricity and spatial frequency of the stimulus. One parameter determines the “meso-mesopic” luminance—the center of the mesopic range, at which rod and cone contributions are balanced. This increases with eccentricity, reflecting an increase in the meso-mesopic luminance from 0.04 scotopic cd/m^{2} at 2° eccentricity to 0.44 scotopic cd/m^{2} at 18°. The second parameter represents the slope of the log–log threshold-versus-intensity curve (TVI curve) for rod vision. This parameter inversely scales the width of the mesopic range and increases only slightly with eccentricity (from 0.73 at 2° to 0.78 for vision at 18° off-axis).

*V*(

*λ*) and

*V*′(

*λ*)) is historically, to a large degree, based on flicker photometry. This definition of the photopic standard observer does, with limitations, agree with a weighted achromatic response of L and M cones.

_{10}, Y

_{10}, Z

_{10}and the scotopic luminance. The weights are a function of the equivalent mesopic adaptation luminance and represent the contribution of three cone components and one rod component to brightness perception. The basis for the weighting functions is heterochromatic brightness matches with a bipartite field at various adaptation levels.

*X*‐model, which is based on reaction times for the detection of targets at 0° and 15° under different illuminations (He, Rea, Bierman, & Bullough, 1997). Their model is based on a parameter

*X*that denotes the relative contribution of photopic luminous efficacy. Based on different visual tasks in nighttime driving like reaction times and chromatic and achromatic contrast thresholds for the detection and recognition of objects, Goodman et al. (2007) proposed the MOVE model for mesopic photometry for the target eccentricity of 10°. Their practical approach assumes a linear combination of

*V*(

*λ*) and

*V*′(

*λ*) to form a mesopic luminous efficiency function, with the weighting factor depending on the photopic and scotopic luminance.

*X*‐model based on reaction times, do not represent achromatic luminance. Furthermore, mesopic perception highly depends on visual field position, since rods and cones and their connecting circuitry with distinct temporal and spatial properties are nonuniformly distributed over the retina. These dependencies and their relationships have not been considered sufficiently in mesopic models so far.

*heterochromatic color grating*can be decomposed into two colored sinusoids changing in time and space with the same temporal and spatial frequency. One sinusoid is generated by modulating the red phosphor and the other by modulating the blue phosphor. The red and the blue sinusoidal gratings are 180° out of phase both in time and space. One of the phosphors is always modulated with the maximum possible range (here the red), while the amplitude of the other phosphor's sinusoid (blue) is varied by the observer until a motion null is perceived. At the minimum motion point, the two phosphors are isoluminant.

*homochromatic luminance grating*or “luminance lure” is generated by modulating all three phosphors around the background level. The luminance grating changes with the same spatial and temporal frequency as the color grating. But, now, the phosphors are varied with no phase offset, thus producing a dark–light neutral gray grating that changes in luminance but not in chromaticity. With the intention of keeping the space average in chromaticity of the whole stimulus constant at the chromaticity of the background, the color of the luminance grating is chosen to match the chromaticity coordinates of the background. The homochromatic luminance grating is offset by 90° to the heterochromatic grating in both space and time (that is, it is in quadrature with the colored grating). The maximum modulation amplitude of the luminance lure is kept constant at 8%.

*x*-axis. Each curve in Figure 2 shows the red and blue phosphor intensities R and B over space at a particular moment in time. All gratings are modulated around the red and blue phosphor intensities of the background

*R*

_{BG}and

*B*

_{BG}. The phosphor intensities are the linearized output voltages of the CRT monitor normalized between 0 and 1. Thus, a red phosphor intensity of 0.2 corresponds to 20% of the maximum CIE luminance the red gun can produce. The two solid curves in Figure 2 are summed to generate the color grating. Note that the amplitude of the red grating is chosen to be maximal. At a background level of

*R*

_{BG}= 0.4 or 0.6, a maximal modulation of 0.4 to both sides would be possible. While the amplitude of the red phosphor Δ

*R*was fixed, the amplitude of the blue phosphor Δ

*B*was varied by the observer. The two striped gratings and a sinusoidal modulation of the green phosphor are superimposed in phase to create the gray luminance grating of the same chromaticity as the background. Hence, this luminance grating varies only in intensity. All sinusoids are counterphasing: Each grating appears alternately in its initial spatial phase and in the opposite spatial phase and at intermediate times becomes uniform when the sinusoidal time course of its contrast crosses zero.

*T*

_{1}, the color grating consisting of red and blue alternating patches is presented and at

*T*

_{2}replaced by the gray luminance grating. Note that the offset from the color grating is one-fourth of the cycle (phase shift of 90°). The rapid switch between the two gratings lets the red patch seem to move toward one of the neighboring gray patches of the luminance grating. If the luminance of the red patch is smaller than that of the blue patch, the red patch will jump toward the darker patch of the luminance grating, meaning, to the left (black arrows). Conversely, when the luminance of the red patch is higher than that of the blue patch, the red patch will jump toward the brighter patch of the luminance grating, meaning, to the right (light gray arrows). Frames 3 and 4 show again the red–blue color grating and the gray luminance grating reversed in contrast (shifted by 180°) compared to frames 1 and 2, respectively. Again, each patch will appear to jump toward the right or left patch of the next frame depending on its luminance amplitude.

*T*

_{1}to

*T*

_{4}represent one temporal cycle of the stimulus and are repeated endlessly. This causes the impression of continuous motion. However, if the sensation luminance of the red and blue patches is equal, there is no preference for either motion direction and the stimulus will not be perceived to move but to flicker.

*T*

_{1}to

*T*

_{4}until at

*T*

_{5}it reverts to the grating shown at

*T*

_{1}. The luminance grating is varied sinusoidally in the same way. When the two counterphasing gratings are combined, the intensity of each phosphor at each pixel varied sinusoidally in time. This creates a smoother impression of motion than is possible with the square-wave stimulus of Figure 3, and it also has the advantage of allowing systematic exploration of the effects of spatial and temporal frequency on the contribution of rods and cones to mesopic luminance.

*x,*e.g., pixels on the monitor. The color grating can be decomposed into a luminance component (achromatic) and a chromaticity component as shown in the second diagram of each row. The chromaticity component is present for any setting of the relative intensity of the red and blue phosphors, since they are always in opposite spatial and temporal phases (black curves in the graph's second column). However, luminance modulation only appears in the color grating if and only if the red and blue phosphors have different amounts of luminance information (cases A and B). Whenever the achromatic component of the blue and red sinusoids cancel each other (case C), the luminance profile of the chromatic grating is uniform (light gray curves in the second column graphs) and produces no motion effect by its interaction with the luminance lure. Only the interaction of the luminance component of the heterochromatic color grating with the homochromatic luminance grating results in the perception of motion.

*πxf*

_{S}) · cos(2

*πtf*

_{T}) and −cos(2

*πxf*

_{S}) · cos(2

*πtf*

_{T}). Here,

*f*

_{S}and

*f*

_{T}denote the spatial frequency and the temporal frequency of the gratings, respectively. Given that the luminance grating is shifted by 90° in time and in space compared with the color gratings, it can be described by the product of the sine over time and the sine over space: sin(2

*πxf*

_{S}) · sin(2

*πtf*

_{T}). From this, the variation of the red phosphor can be expressed as

*P*

_{B}and

*P*

_{R}, respectively, whereas

*m*

_{Lum}is the modulation amplitude of the homochromatic luminance grating (“luminance lure”) and

*m*

_{R}and

*m*

_{B}determine the modulation amplitude of the red and blue sinusoids of the color grating. Here,

*m*

_{R}is fixed at a level that allows maximum modulation of the red phosphor and

*m*

_{B}is varied by the observer. The green phosphor is set to the level of the background and not altered:

*m*

_{Lum}(

*P*

_{R}+

*P*

_{B}) / 2. It has to be emphasized that the amplitude of the luminance grating is fixed at all times with

*m*

_{Lum}= 0.05, whereas the ratio of the amplitudes between the red and the blue components of the color grating is variable.

*P*

_{R}+

*P*

_{B}. Since the two gratings are modulated on the adaptation background, at any time, the space average of the photopic CIE luminance of the stimuli matches the photopic CIE luminance of the background. Therefore, local adaptation causing side effects is avoided.

*x, y*) = (0.34, 0.33)) at seven luminance levels (0.011 cd/m

^{2}, 0.022 cd/m

^{2}, 0.044 cd/m

^{2}, 0.088 cd/m

^{2}, 0.43 cd/m

^{2}, 1.9 cd/m

^{2}, and 42 cd/m

^{2}). The scotopic adaptation condition was realized with a purple background of 0.0024 cd/m

^{2}, because the eew background did not offer a high enough range for modulation of the red phosphor to find a motion null with pure rod vision.

*x, y*≈ 0.37, 0.22) of the same

*S*/

*P*ratio as the equal energy white under a variety of adaptation conditions. Here, the green phosphor was set to 0 for the background and the stimuli. In addition, several colored backgrounds with

*S*/

*P*ratios ranging from 0.7 to 3.4 were used by keeping the photopic luminance constant and altering the scotopic luminance of the background alone. To allow for peripheral and near foveal matches, the radii of the annulus were set to 1°, 2°, 5°, 10°, 14°, or 18°. The given radii are the mean of the inner and outer radii. The relative thickness of the rings was set to 15% of the radius, except for the smallest two stimuli, whose thickness was set to 30% of their radii. This maintains visibility and allows minimum motion settings at dimmer light levels. For each stimulus condition, at least 5 repeated settings were done. For the main part of the experiment, the spatial frequency of the annulus was kept constant at 1 cycle per degree (cpd) for all stimuli sizes by setting the number of windmill segments accordingly. The temporal frequency

*f*

_{T}of the annulus, which is the frequency of contrast reversal of each sinusoid, was kept constant at 2 Hz for the main experiment. A 2-Hz frequency is low enough to avoid appreciable phase lags between rod and cone signals that can complicate the time cancellation between rods and cones (MacLeod & Stockman, 1987), yet high enough to give the impression of a smoothly rotating annulus. An overview of the stimulus parameters and the variables is shown in Table 1.

Independent variables | ||
---|---|---|

Background/adaptation luminance, P _{adapt} | 0.0024–42 photopic cd/m^{2} | |

Equal energy white (x, y) = (0.34, 0.33) or purple (x, y) = (0.37, 0.22) or green (x, y) = (0.31, 0.43), S/P ratio ≈ 1.7–2 | ||

Annulus radius/eccentricity | 1°/2°/5°/10°/14°/18° | |

Temporal frequency, f _{T} | 2 Hz (occasionally 0.5 Hz–4 Hz) | |

Spatial frequency, f _{S} | 1 cpd (occasionally 0.5 cpd–5 cpd) | |

Parameters and conditions | ||

Thickness of annulus | 15% of radius, 30% of radius for 1° and 2° | |

Repetition of each stimulus | 5 or more | |

Subjects | Eight subjects took part in the main experiment, 1 further subject took part only in 2 follow-up experiments, visus ≥0.8, normal color vision, age: 18–36 (4 ♂, 4 ♀) | |

One deuteranomalous observer took part in the bleaching experiment (age 65) |

*x, y*= (0.37, 0.22) and that of the green background were

*x, y*= (0.31, 0.43). A further experiment examined the change of rod contribution with increasing scotopic luminance at constant cone stimulation; here, the photopic luminance of the background was kept constant and its scotopic luminance was varied up to an

*S*/

*P*ratio (scotopic luminance/photopic luminance) of 3.4.

*R*and Δ

*B*) at isoluminance was recorded and transformed into the photopic luminance

*P*(based on

*V*

_{10}(

*λ*)) and scotopic luminance

*S*(based on

*V*′(

*λ*)). A simple linear modeling approach was used to calculate relative rod and cone weights in such a way that the weighted linear sum of photopic and scotopic luminance is constant as shown in the following equation:

*S*and

*P*represent the scotopic and photopic luminance levels of the red and blue amplitudes of the color sinusoid for isoluminance at one adaptation level and eccentricity.

*W*

_{P}′ and

*W*

_{S}′ denote the independent weights for the photopic and scotopic luminance, respectively. For each adaptation level and eccentricity, the weights were iteratively calculated. The sum of the weights

*W*

_{P}′ and

*W*

_{S}′ was normalized to 1 to retrieve the relative weights

*W*

_{P}and

*W*

_{S}. Hence, a relative cone weight

*W*

_{P}of 1 (

*W*

_{S}= 0) represents pure cone vision if

*V*

_{10}(

*λ*) is the underlying sensitivity function, and similarly, a relative rod weight

*W*

_{S}of 1 (

*W*

_{P}= 0) corresponds to pure rod vision as defined by

*V*′(

*λ*).

^{2}, the adjustment was more difficult or for some subjects not feasible at all. The 2° radius data, however, do show an increase of rod activation for lower adaptation levels, though they do not reach a pure scotopic response at the lowest tested level of 0.01 cd/m

^{2}. At 18° off-axis, vision is—according to our simple linear model—entirely dominated by rods at 0.06 cd/m

^{2}, a luminance where the 2° radius isoluminance setting remains closer to photopic than scotopic equality. Below that luminance, the blue sensitivity at 18° increases further to a level that reliably exceeds expectations based on the standard scotopic luminosity curve

*V*′(

*λ*).

^{2}, the average relative cone weights for the two smallest stimuli are slightly above unity. This implies a decreased blue sensitivity within a 1–2° field radius compared to

*V*

_{10}(

*λ*), which is based on a field with a 5° radius and is an expected consequence of the increased density of short-wavelength-absorbing macular pigment in or near the fovea. At this highest luminance level that we ascribe to pure cone vision and also at the dimmest luminance level of 0.002 cd/m

^{2}that we can safely attribute to pure rod vision, the relative weights show a small but significant dependence on eccentricity. The topmost and lowest curves in Figure 6 show this effect: a consistent increase of sensitivity to short wavelength with increasing eccentricity. Under photopic conditions, this sensitivity profile might be caused by rod contribution to luminance. We tested for possible rod intrusion by bleaching the lower part of the visual field with a 50,000 cd/m

^{2}blue light (generated by a BIGMAX Projector) for 10 s. Motion nulls were set subsequent to bleaching with an annulus of 6° radius modulated on a 42 cd/m

^{2}gray background. One deuteranomalous observer was tested. The average isoluminance settings of the bleached and unbleached conditions showed no differences 1 and 20 min after the bleach and also match isoluminance settings of that subject without bleaching. The results support the assumption that the nulling task is accomplished by a pure cone response at the highest adaptation luminance of 42 cd/m

^{2}, whereas the rod response becomes insignificant.

*M*shown in Figure 7 in units of scotopic cd/m

^{2}is strongly dependent on eccentricity. It increases from 0.04 sc cd/m

^{2}at 2° eccentricity to 0.13 sc cd/m

^{2}at 5° and 0.44 sc cd/m

^{2}at 18° (see Table 2), demonstrating the extent to which the position of the stimulus influences perception under mesopic adaptation levels. The strong effect is not unexpected in view of the distinct distribution of rods and cones across the retina. Central vision is determined by the cones alone also at mesopic light levels due to the absence of rods in the fovea, whereas with eccentricity, the relative amount of rods in the retina increases, so does their influence on vision.

Radius of the annulus in degrees | k | M (in sc cd/m^{2}) | RMSE |
---|---|---|---|

2 | 0.73 | 0.04 | 0.028 |

5 | 0.75 | 0.13 | 0.030 |

10 | 0.72 | 0.24 | 0.018 |

14 | 0.76 | 0.35 | 0.023 |

18 | 0.78 | 0.44 | 0.019 |

_{1964}standard observer and the psychophysically measured sensitivities that are evident as relative cone weights above 1 and below 0 in Figure 5 make it difficult to describe the results mathematically in terms of the standard observer. Therefore, the scotopic and photopic luminances were redefined as “sensation luminance” to accommodate the scotopic and photopic averaged results of all observers. Sensation luminance is a term Kaiser (1988) used to define luminance that is based on the individual spectral sensitivity. For quantitative analysis, we accommodate the deviations from the photopic and scotopic standard observers by considering observer-specific scotopic and photopic sensation luminances,

*S*′ and

*P*′ as described more fully in 1.

*S*′, with

*k,*the slope of the rod TVI curve (log of increment threshold intensity vs. log of background intensity), and the meso-mesopic luminance

*M*as parameters. The rationale for this model is that, in the mesopic range, cone sensitivity changes little with illumination level, whereas rod sensitivity undergoes changes that are reflected in the variation of incremental threshold with intensity (see 2 for details). Specifically, the relative cone weight

*W*

_{P}is given by

*S*has here been replaced by the scotopic sensation luminance

*S*′. The parameters

*M*and

*k*were optimized iteratively by minimizing the root mean square error (RMSE) between the

*W*

_{P}vs. log(

*S*′) curve and the fitted curve for each annulus diameter. The fitted curves for the annulus radii between 1° and 18° are shown in Figure 8. The variation of the cone contribution to luminance with light level is well described if the exponent

*k*takes values between 0.72 and 0.78. This is in gratifying agreement with rod TVI slope values found in detection of flashes (Aguilar & Stiles, 1954; Barlow, 1957; Sharpe, Fach, Nordby, & Stockman, 1989).

*k, M,*and RMSE.

*S*/

*P*ratio. With this choice of the red/blue balance, a purple and a green field of the same photopic luminance are, by definition, equal in scotopic luminance as well. If now the scotopic and photopic spectral luminosities are the only quantities that determine mesopic luminance information, the equal energy white and the two colored fields should elicit the same mesopic response. Otherwise, however, this is not generally to be expected, since the excitations of L and M cones are quite different for the purple and green fields.

*S*/

*P*ratio the same (Figure 9). Next, we investigated how the relative weighting of rod and cone inputs for mesopic luminance depends separately on the photopic and scotopic luminance of the stimulus. The scotopic luminance of the background field was varied by varying proportions of red and blue, while the photopic luminance was held constant at each of a range of values. The influence of various

*S*/

*P*ratios ranging from 0.7 to 3.4 on the isoluminance ratio was tested with minimum motion settings for five adaptation levels between 0.008 and 24 photopic cd/m

^{2}. By systematically varying the scotopic luminance

*S*and keeping the photopic luminance

*P*constant at one of the five levels, the results should reflect the consequences of rod adaptation (though an influence of S cone excitation is not excluded

*a priori*). Figures 10–12 show the change of the relative cone weights over a range of

*S*/

*P*ratios for the two observers, with the

*P*value fixed within each figure; thus, the horizontal axis can equivalently represent either the

*S*/

*P*ratio or the scotopic luminance of the field (as a multiple of the fixed photopic luminance). An increase of relative cone weights for increasing

*S*/

*P*ratio is evident for the three mesopic adaptation levels of 0.027 photopic cd/m

^{2}, 0.054 photopic cd/m

^{2}, and 0.29 photopic cd/m

^{2}(Figures 10 and 11).

^{2}, rod influence is miniscule, except at the largest eccentricity tested (14°), where a clear deviation from photopic sensitivity is evident for reddish background fields with an

*S*/

*P*ratio of 1 or less (see Figure 12). The nulls for the 2° stimulus at this relatively high luminance, on the other hand, show slightly less short-wavelength sensitivity than expected for a photopic match, hence a “relative cone weight” in excess of unity. This is not unexpected, since our calculation of the cone weights is based on the large field spectral sensitivity curve

*V*

_{10}(

*λ*) and the effective density of macular pigment will generate such a deviation.

*S*/

*P*ratio is related to the slope of the rod TVI curve through parameter

*k*. Table 3 shows the values of parameter

*k*for a best fit to the weights in Figures 10 and 11 (see 2 for details). While the relative cone weight at mesopic adaptation levels increases with increasing

*S*/

*P*ratio, the weights in the photopic condition are close to 1 and do not change substantially. Thus, the rod TVI slope

*k*was constrained only by the gradients of the curves in Figures 10 and 11 for the mesopic conditions.

P _{adapt} (cd/m^{2}) | Radius (deg) | TVI slope k |
---|---|---|

Observer 1 | ||

0.027 | 2 | 1.48 |

0.027 | 5 | 1.75 |

0.29 | 2 | 1.16 |

0.29 | 5 | 1.22 |

0.29 | 14 | 1.68 |

Observer 9 | ||

0.054 | 2 | 1.07 |

0.29 | 2 | 1.00 |

0.29 | 5 | 1.51 |

0.29 | 14 | 1.19 |

*k*values in Table 3 are generally greater than 1, hence greater than the slope of the rod TVI curve (Aguilar & Stiles, 1954)—and also steeper than was suggested by the results of Figures 5 and 8 where stimulus intensity rather than color was varied. Thus, while the model of Equation 5 and 2, with a

*k*value consistent with the rod TVI curve, gives a good account of the variation in cone weight with luminance (Figures 5–9), such a model underestimates the effects of variation in adapting color when the adapting

*S*/

*P*ratio varies (Figures 10–12). Evidently, the relative contribution of rods and cones to luminance is particularly strongly influenced by the relative balance of photopic and scotopic excitations by the background. Exactly analogous results have been reported for the relative contribution of L and M cones to photopic luminance: There, changes in background color selectively suppress the more strongly adapted of the two cone systems to a much greater extent than would be expected from the effects of varying background intensity (Eisner & Macleod, 1981).

^{2}. At 5 Hz and slightly higher frequencies, the stimuli appeared to flicker at all relative luminance settings. This made the nulling of motion difficult and is expected if phase-shifted rod and cone signals are being combined at this temporal frequency.

^{2}the cone weight increased sharply at the highest spatial frequency, showing that even in this very dim condition the cones play a substantial role at high spatial frequencies. The relative increase of cone weights for the intermediate mesopic light level of 0.09 cd/m

^{2}is similar for near foveal and peripheral stimuli (Figure 15). At photopic adaptation levels, the cone contribution is independent of spatial frequency. In this frequency range, the effects of chromatic aberration on the motion nulls are small (Cavanagh et al., 1987; Curcio et al., 1991).

_{10}unit reduction of luminance. The present results are somewhat different: At photopic light levels, the blue sensitivity measured as the isoluminance ratio Δ

*P*

_{R}/Δ

*P*

_{B}increases linearly with the log of the radius in degrees, but as in Anstis' results at lower adaptation levels, the blue sensitivity increases more rapidly with the radius.

*V*

_{10}(

*λ*) spectral luminosity function for eccentricities between 2° and 5°. For more peripheral photopic stimuli, sensitivity to blue increases slightly beyond the sensitivity described by

*V*

_{10}(

*λ*). For the largest annulus of 18° radius, the scotopic isoluminance based on

*V*′(

*λ*) was reached at 0.06 cd/m

^{2}, decreasing to 0.008 cd/m

^{2}for stimuli with a radius of 5° (Figure 5), but negative relative cone weights, implying a further slight increase in short-wavelength sensitivity, were obtained at lower luminances. These differences in cone weights between the 5° radius and the larger annuli at the lowest luminance level are surprising, given that, at this luminance, vision is determined by rods alone and that the sensitivity of the rods is constant over the peripheral visual field. The CIE

*V*′(

*λ*) function is based on data collected at 8° eccentricity by Wald (1945) and with a 10° radius bipartite field by Crawford (1949). One might expect agreement with the present data for a field radius between 5° and 10°. However, the results suggest that rod sensitivity in the peripheral retina favors short wavelengths to a greater degree than expected on the basis of

*V*′(

*λ*). Likewise, the finding that the isoluminance ratio at the highest tested luminance is not independent of eccentricity is surprising since one would expect this invariance for a pure cone response. Instead, the subjects' sensitivity for short wavelength increased slightly between 5° and 18° off-axis (Figure 6). This effect is small in comparison to the large influence of eccentricity at mesopic light level, but it is systematic.

^{2}. Rods begin to saturate around 2.0 log

_{10}scotopic trolands (sc td; ≈3 cd/m

^{2}) but are not completely saturated until 120–300 cd/m

^{2}(Aguilar & Stiles, 1954; Hayhoe, MacLeod, & Bruch, 1976; Stockman & Sharpe, 2006). We tested for possible rod intrusion by bleaching the lower part of the visual field before acquiring motion nulls in a deuteranomalous observer and found no differences between bleached and unbleached isoluminance settings. This rules out a measurable rod contribution to the nulling task and suggests a pure cone response at the highest adaptation luminance employed here.

^{2}between 5° and 18° off-axis, a peak macular pigment optical density (MPOD) of 0.25 ± 0.06 is necessary for the eight subjects that took part in the main experiment. From 1° to 18°, the shift would require an average change in peak MPOD of 0.89 ± 0.38. Under the scotopic adaptation luminance of 0.002 cd/m

^{2}, the shift between 5° and 18° off-axis can be explained by an MPOD difference of 0.24 ± 0.07, a value substantially higher than found by Stockman et al. (1993) for a full 10° field. (These necessary changes in macular pigment optical density were calculated with the macular pigment absorbance spectrum from Bone, Landrum, and Cains (1992) and Stockman and Sharpe (2000) for a 2° visual field.)

*M,*which determines the position of the curves of Figure 8 on the log intensity axis, and the exponent

*k,*which determines the rapidity of the mesopic transition reflected in the horizontal scale of those curves. The exponent

*k*agreed well with estimated slopes of the rod TVI curves

*k*from detection threshold experiments (Barlow, 1957; Blackwell, 1946; Sharpe, Fach, & Stockman, 1992). Part of the rationale for this very simple model is that the cone incremental threshold is approximately constant at light levels below 40 td. Therefore, the relative cone weight is set by the light level defined in terms of scotopic (rod-weighted) luminance, with at most a minor influence of photopic luminance.

*S*/

*P*ratio, which excite rods strongly, decrease the effectiveness of modulated rod stimulation by the test stimulus. Like the effect of changing luminance, this is qualitatively expected from the near-Weber behavior of the rod system in adaptation at mesopic levels. However, the results for changes in background color required

*k*values above 1, higher than those estimated from the main experiment, which were below 1 (Table 2), in agreement with direct TVI curve measurements in the earlier work.

*S*/

*P*ratio increases the light level for rods without any increase for cones, and this could reduce rod sensitivity in two ways: (1) by the independent action of the adapting stimulus on the rod system, an effect reflected in the slope of the rod TVI curve seen when only the background intensity changes, and (2) by a change in the balance of stimulation in favor of cones, which allows the cones to further suppress the rod signals when the adapting stimulus has a spectral composition such that it excites cones relatively strongly.

*incremental additivity,*which is the only kind of additivity attainable in a mesopic photometric system (Rea et al., 2004). The modulations of rod and cone excitation around the adapting level combine to generate a luminance contrast that is the weighted sum of photopic and scotopic luminance contrasts and is zero at the motion null. In this way, the model can very simply specify the mesopic luminance contrast of any mesopic test stimulus once the state of adaptation is given.

*k*(that determines the width of the mesopic log intensity range) with the corresponding value from the TVI curve slope.

*k*needed to account for the color data. To address this, the term

*S*

^{−k }in Equation 5 could be joined by a color-dependent term (

*P*/

*S*)

^{ j }, where

*j*is the difference between

*k*and the exponent that fits the color data. Analogously in the photopic case, the suppressive effects of chromatic adaptation on photopic flicker photometric sensitivity exceed expectations based on Weber's law and require a similar added factor to model them (Eisner & Macleod, 1981). The added factor will often have a relatively minor effect in practice, since environmental variation in luminance so greatly exceeds variation in

*S*/

*P*.

*M*alone; this horizontally translates the fixed curve shape from Equation 5 along the log intensity axis.

*S*/

*P*ratio

*S*/

*P*ratio were used for adaptation. In this case (unlike the previous case, where background color changes led to variation in

*S*at constant

*P*), the relative photopic luminance weight was approximately invariant with background color. This supports the notion of a mesopic channel comprised of the rod signal and a photopic luminance signal formed by the summed L and M cone activations. The invariance with background color under these limited conditions is not conclusive evidence against the participation of opponent channels or of S cones in the motion nulls, but precise tests in prior experiments at photopic levels (Cavanagh et al., 1987) have not supported the hypothesis of S cone participation. Altogether, the similar cone weights for the different adaptation fields support the interpretation of a mesopic luminance based on rod and cone achromatic luminance information that determine minimum motion settings under all adaptation conditions. The results can be described as a linear weighted sum of scotopic and photopic luminance based on the scotopic luminosity function

*V*′(

*λ*) and the photopic 10° luminosity function

*V*

_{10}(

*λ*).

^{2}, for example, the data of Figure 15 show a change in relative cone weight from 0.29 at 1 cpd to 0.7 at 4 cpd. This means that the ratio of cone sensitivity to rod sensitivity changes from 0.29/(1 − 0.29) = 0.41 to 0.7/(1 − 0.7) = 2.33, thus by a factor 5.7. Likewise in D'Zmura and Lennie (1986), cone sensitivity (at 2.9 td) was roughly constant between 1 and 4 cpd, while rod sensitivity dropped by a factor of about 4. These results imply that differences in luminance and retinal position do not fully account for the difference between photopic and scotopic contrast sensitivities: The contrast sensitivities of rods and cones differ considerably even when tested at the same retinal position and luminance as we have done.

^{2}in Figure 16, which are the ratios

*P*

_{r}/

*P*

_{b}of the photopic luminance of the red phosphor and the photopic luminance of the blue phosphor at motion null, it appears that they may fall into two groups with distinct photopic sensitivity. Especially, the isoluminance ratios for the 18° and 14° annulus radii seem to separate in two groups (with average isoluminance ratios of 1.3 vs. 1.7 and 1.2 vs. 1.6, respectively). A plausible idea is that this disparity might be due to variations in L cone pigments. The two polymorphic variants of L cone pigment incorporating alanine vs. serine result in a shift of a few nanometers of the absorbance spectrum. The shift has been estimated as 2.7 nm (Sharpe et al., 1998) or perhaps more (4.3 nm, Merbs & Nathans, 1992; 7 nm, Asenjo, Rim, & Oprian, 1994). This hypothesis does not, however, survive quantitative scrutiny. Let us assume a shift of 3 nm in L cone peak excitation between observers with serine and alanine forms of the normal L cone pigment. The relative excitation of L cones (which are the main source of photopic luminance and of bimodal variation in normal color vision) by the blue and red phosphors changes by about 3% per 1-nm shift in the peak absorption (Golz & MacLeod, 2003, Table 6). This suggests that the serine/alanine polymorphism of the L cone pigment, with its attendant shift in wavelength of peak absorption by about 3 nm, will change the photopic relative luminance of the red and blue phosphors by about 10%. The individual variation revealed at large eccentricities in Figure 16 is of much greater magnitude and, therefore, probably has a different origin, such as either continuous or discrete individual variation in the relative contributions of the L and M cones to photopic luminance.

^{2}and 2 cd/m

^{2}showed, for our observers, diameters between 3 and 5 mm, respectively, which is typical of values in the literature (e.g., Bouma, 1965; Trezona, 1983). Variations in pupil size for different adaptation conditions might influence the relative stimulation of rods and cones. However, within a large luminance range, the relation between pupil area and log luminance is roughly linear for large field sizes (Bouma, 1965; Le Grand, 1968; Stanley & Davies, 1995). The 34° × 44° field used here corresponds to a circular field with a diameter of 43.6° in area. Extrapolation of earlier data for a 43.6° field shows that the relation between pupil area and log luminance is largely linear between 40 cd/m

^{2}and 0.002 cd/m

^{2}—the luminance range used here (Bouma, 1965; Stanley & Davies, 1995; Trezona, 1983). Pupil constriction will affect the relative receptor weights only slightly: If the Stiles–Crawford effects of rods and cones are neglected, the ratio between the scotopic and photopic luminances is the same as the ratio between scotopic and photopic trolands. Hence, the main effect of pupil size is to slightly compress the range of log retinal illuminance relative to the range of external luminance. The value of the exponent

*k*in our model should, thereby, be slightly increased when applied to retinal illuminance rather than external luminance, but this adjustment is less than 10%.

^{2}. Such an arbitrary adjustment can encompass both the effects of possible individual factors such as variation in macular pigment density and the effects of calibration errors or uncertainties that we discuss in the Discussion section. The following equation defines the sensation luminance

*P*′ as the sum of the weighted luminance contribution of all three phosphors:

*P*

_{R},

*P*

_{G}, and

*P*

_{B}are the maximal photopic luminances the three phosphors can produce (for the standard observer);

*r, g,*and

*b*denote the phosphor intensities that are the linearized output voltages of the CRT monitor normalized between 0 and 1. The observer-specific weights for the red and blue phosphors are

*ω*

_{R,phot}and

*ω*

_{B,phot}, respectively. The red and blue phosphor sensation luminances, relative to that of the green phosphor, are

*rω*

_{R,phot}and

*bω*

_{B,phot}, respectively, as compared with simply

*r*and

*b*for the standard observer. The phosphor weights for the photopic sensation luminance

*P*′, relative to those of the standard 10° observer, were in the range

*ω*

_{R,phot}= 0.87–0.97 and

*ω*

_{B,phot}= 1.07–1.31. Scotopic sensation luminance

*S*′ was defined in the same way as

*P*′ (now allowing for scotopic isoluminance at 0.0024 cd/m

^{2}) with weights of

*ω*

_{R,scot}= 0.75–0.90 and

*ω*

_{B,scot}= 1.1–1.26 (for radii ≥ 5°). The given range of weights results from multiple tested stimuli radii, where the larger stimuli are associated with the lower weights of the red phosphor and the higher weights for the blue phosphor. For the radii of 1° and 2°, the weights and the fitted curves (Figure 8) are based on the scotopic (

*V*′(

*λ*)) and photopic (

*V*

_{10}(

*λ*)) luminance and not on sensation luminance.

*S*′ and

*P*′, can be described with a sigmoid function.

- Under mesopic and low photopic light levels, rods work nearly in accordance with a power law generalization of Weber's law, that is, incremental rod detection threshold Δ*
*S*is proportional to the*k*power of background intensity*S,*with*k*not far below 1 where a value of 1 would imply conformity with Weber's law (Wyszecki & Stiles, 1982). Thus$ \Delta * S\u2062\u223c S k .$(B1) - The cone incremental threshold is treated as constant at adaptation conditions below 40 td (∼2.5 cd/m
^{2}). Accordingly, we assume that the effectiveness of cone stimulation, which we represent by an independent cone weight*W*_{P}′, will be constant and that variations in the relative influence of rods and cones reflect variation in the effectiveness of rod stimulation, which we represent by an independent rod weight*W*_{S}′. - The proportional relation between rod incremental threshold and rod background intensity to the power of
*k*is also valid for modest multiples of rod thresholds (as found by MacLeod, 1974). Hence, equal multiples of Δ**S*are equal in the luminance contribution from the rods, and the independent (not normalized) weight for the rods*W*_{S}′ will be inversely proportional to the rod threshold from Equation B1:$ W S \u2032 \u223c S \u2212 k .$(B2)

*W*

_{P}′ is constant and that rods and cones do not interact, this relation leads to

*W*

_{P}is the relative cone weight defined as

*W*

_{P}=

*W*

_{P}′ / (

*W*

_{P}′ +

*W*

_{S}′) = 1 −

*W*

_{S}so that the relative cone and rod weights

*W*

_{P}and

*W*

_{S}sum to 1, and

*M*is the meso-mesopic luminance in scotopic cd/m

^{2}. The scotopic standard observer adapting luminance

*S*has here been replaced by its sensation luminance counterpart

*S*′.

*S*/

*P*ratio is expected to result in a decrease in rod sensitivity (at a constant photopic adapting luminance) and a correspondingly reduced relative weight for scotopic intensity differences. Likewise, a low

*S*/

*P*ratio, producing a reddish adapting field, will lead to an increased relative rod weight. From the variation in relative cone weight introduced by such variations in background color (Figures 10 and 11), Equation B3 was used to estimate

*k*in just the same way as when background luminance varies. But, as noted, the values of

*k*obtained with variation in color or the adapting stimulus are greater than those obtained for variation in adapting intensity. We note that for relative rod weights close to 1 the independent rod weight is large relative to the independent cone weight. Hence, a doubling of the independent rod weight due to a decreasing

*S*/

*P*ratio will only lead to a small change in the photopic relative weight. The same is true if the relative rod weight is close to 0 and the relative cone weight is close to 1. In these cases, a rather large change of the smaller independent weight will not change the relative weight in the same degree. As a consequence of this compression of the relative weights, the TVI exponent

*k*could not be accurately estimated from weights in close proximity to 0 or 1. The best-fitting values of

*k*in Table 3 are, therefore, mainly constrained by the slopes of the curves of Figures 10 and 11 where neither rods nor cones predominate.