Mesopic luminance assessed with minimum motion photometry

Abstract

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/m² at 2° eccentricity to 0.44 scotopic cd/m² 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).

Introduction

Mesopic vision is characterized by the joint stimulation of rods and cones. The mutual activity and interactions between the two different kinds of receptors make twilight vision highly complex and the perception-based evaluation and luminance photometry particularly challenging. Determining luminance as defined by the CIE photopic and scotopic standard observers (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. 

Describing the achromatic luminance under mesopic conditions is rendered difficult by the complex temporal behavior of joint rod and cone vision (Stockman & Sharpe, 2006). A mesopic equivalent of luminance with flicker photometry is particularly difficult to accomplish because the temporal differences between the rod and cone pathways lead to signal cancellations and enhancements (MacLeod, 1972; Stockman & Sharpe, 2006; Vienot & Chiron, 1992). There have been several attempts to determine a mesopic equivalent of luminance with differently colored patches that were matched for brightness (e.g., Ikeda & Shimozono, 1981; Kokoschka, 1972, 1980; Palmer, 1966, 1967, 1968; Sagawa & Takeichi, 1992; Trezona, 1991) or with detection and recognition thresholds of colored stimuli (Goodman et al., 2007). These models define a mesopic equivalent as a sum of weighted rod and cone components that converge to the photopic standard observer at high light levels and to the scotopic standard observer at scotopic light levels. 

Palmer (1966, 1967) measured sensitivity curves by directly matching the brightness within a bisected circular field under a variety of adaptation levels. He proposed a mesopic unit as a simple nonlinear combination of the weighted 10° photopic and scotopic luminance, which involves one parameter that depends on field size (Palmer, 1968). Ikeda et al. quantified mesopic brightness as a weighted sum of the logs of the scotopic and photopic luminous efficiency functions defined by direct brightness matching with the weighting factors defined by the adaptation level (Ikeda & Shimozono, 1981; Sagawa & Takeichi, 1992; Yaguchi & Ikeda, 1984). Kokoschka (1972, 1980) and similarly also Trezona (1991) propose a four-component model as a linear weighted combination of the CIE 10° tristimulus values X₁₀, Y₁₀, Z₁₀ 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. 

Two approaches that intend to provide a measure to assess driving performance under several lighting conditions were proposed. Rea, Bullough, Freyssinier, and Bierman (2004) proposed the 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. 

Heterochromatic brightness matches and detection and recognition thresholds are known to produce spectral sensitivity curves divergent from flicker photometry because these methods comprise signals of opponent color pathways. Hence, the proposed models, with the exception of the 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. 

Here, we adopt a motion nulling technique that relies on information of the achromatic luminance channel and omits signals from the opponent color channels. It has been suggested that motion perception is mediated by the luminance pathway and that color does not contribute to motion (Anstis, 1970; Ramachandran & Gregory, 1978), though this proposal has been qualified with the recognition of at least a minor chromatic contribution to motion sensitivity (e.g., Cavanagh & Anstis, 1991; Cavanagh & Favreau, 1985; Derrington & Badcock, 1985). Although it is possible to perceive the motion of isoluminant stimuli, this is generally achieved by “third-order” processes distinct from the “first-order” motion processing system that determines minimum motion nulls (Lu, Lesmes, & Sperling, 1999). 

The method of nulling the apparent motion of a stimulus composed of homochromatic and heterochromatic sinusoids has been used to determine individual isoluminance ratios between two colors under photopic conditions (Anstis & Cavanagh, 1983; Cavanagh, MacLeod, & Anstis, 1987; Kaiser, Vimal, Cowan, & Hibino, 1989). Cavanagh et al. (1987) suggest that the mechanisms used for motion nulling respond to the linear sum of M and L cone signals only and, hence, are a measure of luminance resembling flicker photometry in this respect (Eisner & MacLeod, 1980; Ripamonti, Woo, Crowther, & Stockman, 2008). However, Cavanagh and Anstis (1991) found a contribution of the red–green chromatic channel, albeit small, which decreased further with temporal and spatial frequency. The critical role for luminance is supported also by the high degree of additivity that was found in minimum motion data. The slight additivity failures of this method are within the same range of additivity failures of flicker photometry (Kaiser et al., 1989). Thus, minimum motion and flicker photometric measures of sensitivity depend both on the summed activation of L and M cones under conditions where rods are excluded by the use of relatively high light levels or small fields. Thus, even though the perception of motion in our stimulus is achieved by phase-shifted colored gratings, the results of the present study can reasonably be presumed to reflect the luminance inputs to first-order motion mechanism. A further key advantage of minimum motion photometry is the ability to set motion nulls at low temporal frequencies where phase differences are small and destructive interferences between receptor pathways are avoided. 

Earlier findings with the method of minimum motion were obtained under photopic conditions and therefore reflect the response of cone vision. The purpose of the present study is to determine the isoluminance ratio for photopic, mesopic, and scotopic adaptation levels at various eccentricities, spatial frequencies and temporal frequencies with the focus on determining the relative contribution of rods and cones to the achromatic luminance channel. The results are fit with a very simple “scotopic exponent” model in which the transition from rod to cone vision in the mesopic range is related to the luminance-dependent incremental sensitivity of the rod system. The model follows the tradition of representing mesopic luminance as a sum of photopic and scotopic luminance with adaptation-dependent weights, so that mesopic luminance is linearly related to the stimulus intensities under any one state of adaptation. 

Methods

The minimum motion stimulus

The minimum motion stimulus used here is a windmill-like colored annulus (Figure 1) that was presented on a CRT monitor. 

Figure 1

 

Minimum motion stimulus composed of red and blue sinusoidal waveforms on a uniform background.

The annulus is composed of two temporally and spatially varying sinusoidal gratings: A heterochromatic color grating and a homochromatic luminance grating, which are both modulated on the uniform background. The heterochromatic colored grating is formed by spatially interleaving two differently colored sine-wave components, a test and a reference, in opposite spatial phase. Depending on the relative sensation luminance information of the two components of the chromatic grating, the stimulus appears to rotate clockwise or counterclockwise. When the two luminance amplitudes match, the annulus will be perceived as neither moving rightward nor leftward; it will appear flickering. This state is called the motion null. 

In the following paragraphs, the composition of the stimulus will be discussed in more detail using a blue and red colored annulus as an example. 

The 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. 

The 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%. 

Figure 2 depicts the described sinusoids that make up the annular test field. Angular position within the annulus is the relevant spatial coordinate and is shown by the 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 ΔR was fixed, the amplitude of the blue phosphor ΔB 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. 

Figure 2

 

The red and blue phosphor intensity, modulated in space (x) around the background intensity R _BG and B _BG, are shown. In addition, the sinusoids alternate in time around their space and time average intensity (=background intensity), hence every half time cycle the contrast reverses. The gratings change in time and space with the same temporal and spatial frequency. The sum of the striped gratings produces a homochromatic luminance grating that changes in intensity but not in chromaticity. The two solid gratings are offset from each other by 180° and form the heterochromatic grating.

With a simplified version of the stimulus as schematized in Figure 3, the perception of motion will be explained. For simplicity, the sinusoids are here replaced by square waves. Likewise, for illustrative purposes, Figure 3 depicts one temporal cycle decomposed into 4 frames that are shown separated vertically and not superimposed as for the stimuli. In frame 1 at 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. 

Figure 3

 

One time cycle of a simplified version of the stimulus, decomposed in four time frames T ₁–T ₄. At T ₁ and T ₃, the heterochromatic color grating (here depicted as square wave) is presented followed by the homochromatic luminance grating at T ₂ and T ₄ offset by a quarter of a spatial period. Between T ₁ and T ₃ and also between T ₂ and T ₄, the signals undergo a periodical contrast reversal, which is repeated in subsequent periods. The apparent motion is leftward when the red patch of the color grating is dimmer than the blue patch. As a consequence, the red patch seems to “move” toward the left, dimmer patch of the luminance grating (dark arrows). An apparent rightward motion is perceived when the red patch is brighter than the blue patch and therefore “moves” toward the brighter patch of the luminance grating (gray arrows). (Figure adapted from Cavanagh et al., 1987.)

For the windmill stimulus of the experiment, the spatial square waves were replaced by spatial sinusoids, and the temporal modulation was introduced by periodically reversing the contrast of the color grating with a sinusoidal modulation in time from 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. 

Figure 4 illustrates how the stimulus can be decomposed into its chromatic and achromatic components. In each inset, the abscissa shows the space dimension 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. 

Figure 4

 

Composition of the stimulus: The graphs express (in the left panels) the modulations of the red and blue intensities over space and (in the center panel) their decomposition in chromatic (black curve) and achromatic luminance (gray curve) information. The change over time is suggested by the black arrows. Case A: The blue color grating is brighter than the red—the interaction of the resulting achromatic information of the color grating and the subsequently appearing luminance grating (right panels) leads to perceived leftward motion. Case B: The red color grating is brighter than the blue—the interaction of the resulting achromatic information of the color grating and the luminance grating leads to perceived rightward motion. Case C: The red and blue gratings contain the same luminance information, which is canceled out, leading to motion null. (Graphs are adapted from Cavanagh et al., 1987.)

A spatiotemporal grating can be described by the product of sinusoids in time and space. Since there is a 180° phase shift between the two colored gratings as shown in Figures 2 and 4, the basic components of the red and blue color gratings can be expressed by cos(2πxf _S) · cos(2πtf _T) and −cos(2πxf _S) · cos(2πtf _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(2πxf _S) · sin(2πtf _T). From this, the variation of the red phosphor can be expressed as 

$R(x,t)=P_{\mathrm{R}}+\frac{1}{2}{P}_{\mathrm{R}}·m_{\mathrm{R}}·\cos (2\pi x{f}_{\mathrm{S}})·\cos (2\pi t{f}_{\mathrm{T}})+\frac{1}{2}{P}_{\mathrm{R}}·m_{\mathrm{L}\mathrm{u}\mathrm{m}}·\sin (2\pi x{f}_{\mathrm{S}})·\sin (2\pi t{f}_{\mathrm{T}}).$

(1)

Likewise, for the blue phosphor: 

$B(x,t)=P_{\mathrm{B}}-\frac{1}{2}{P}_{\mathrm{B}}·m_{\mathrm{B}}·\cos (2\pi x{f}_{\mathrm{S}})·\cos (2\pi t{f}_{\mathrm{T}})+\frac{1}{2}{P}_{\mathrm{B}}·m_{\mathrm{L}\mathrm{u}\mathrm{m}}·\sin (2\pi x{f}_{\mathrm{S}})·\sin (2\pi t{f}_{\mathrm{T}}).$

(2)

The luminance of the background generated by the blue and red phosphors is denoted by 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: 

$G(x,t)=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{.}=P_{\mathrm{G}}.$

(3)

 

In Equations 1 and 2, the sinusoidal terms generate the luminance grating with an amplitude of 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. 

The space- and time-averaged photometric luminances of the annulus for each phosphor in the above equations are equal to the background luminance levels, 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. 

Experimental design

As mentioned in the last section, our minimum motion stimulus created chromatic modulation using the red and blue phosphors of a calibrated CRT monitor (44° × 34°). This ensures maximum visibility under photopic and scotopic conditions since the scotopic and photopic systems are most selectively responsive to the red and blue phosphors. 

To provide background luminances ranging from photopic to scotopic levels, neutral density Lee filters with average optical densities of 0.6 (2 stops), 0.9 (3 stops), and 1.2 (4 stops) and their combinations were set in front of the monitor screen. The output of the CRT monitor was calibrated to linearize the luminance output of each phosphor. In addition, the spectral radiance of the monitor guns as well as the filter spectral transmissions were measured and taken as basis for the calculation of the scotopic and photopic luminances. The monitor was controlled by a Bits++ graphics card from Cambridge Research Systems, which generates stimuli with a 14-bit precision in intensity per color. 

The uniform background was set to an equal energy white (eew, (x, y) = (0.34, 0.33)) at seven luminance levels (0.011 cd/m², 0.022 cd/m², 0.044 cd/m², 0.088 cd/m², 0.43 cd/m², 1.9 cd/m², and 42 cd/m²). The scotopic adaptation condition was realized with a purple background of 0.0024 cd/m², 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. 

In addition to the equal energy white background, two subjects were tested with a purple background (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. 

Table 1

 

Overview of the parameters and conditions.

Independent variables
Background/adaptation luminance, P adapt	0.0024–42 photopic cd/m2
	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)

Eight subjects (age: 18–36, 4 males, 4 females) took part in the experiment. All had at least a visual acuity (visus) of ≥0.8 (20/25) and normal color vision. Two additional subjects took part in the supplementary experiments. 

The duration of adaptation varied from 5 min for the photopic condition to 45 min for the dimmest condition. During dark adaptation, the observers got instructions and got familiar with the stimulus and the adjustment process. The observer's head was positioned at a distance of 40 cm from the screen in a chin rest. The subjects were asked to fixate a fixation cross in the middle of the annulus and to move the mouse until a motion null is found. The stimuli at one adaptation level were shown in random order with 5 repetitions. The data collection for all adaptation levels was divided in 4 sessions spread over several days. In each session, the data for 1–3 adaptation levels were collected. 

In the supplementary experiments, the spatial and temporal frequency were varied between 0.5 and 5 cpd and 0.5 and 4 Hz, respectively. In addition to the equal energy gray background a purple and a green background of the same photometric scotopic to photopic luminance ratio were used. The chromaticity coordinates of the purple background were 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. 

Presentation of the results and calculation of relative receptor weights

The ratio between the amplitude of the red and blue phosphors of the color grating (ΔR and ΔB) 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: 

$\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{.}=S·W_{\mathrm{S}}^{\prime }+P·W_{\mathrm{P}}^{\prime }.$

(4)

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). 

Results

Effect of adaptation level and eccentricity

The average relative cone weights with standard deviations of all subjects are plotted as a function of photopic adaptation luminance and the stimulus radius in Figures 5 and 6, respectively, with the other independent variable as a parameter in each case. 

Figure 5

 

Relative cone weights for increasing adaptation luminance. Parameter is the radius of the annulus. The error bars subtend twice the standard deviation of the individual mean values for each of the 8 subjects.

Figure 6

 

Relative cone weights vs. annulus radius. The parameter is the adaptation luminance and the error bars subtend twice the standard deviation of the individual mean values for each of the 8 subjects.

The graphs show the continuous decline of the cone weight for lower adaptation levels. It is evident that the more eccentric the stimuli, the higher the light level up to which the rods dominate vision. An exception is the very small stimulus with a radius of 1°, where the cones define vision alone at all levels. Here, sensitivity does not change considerably over the tested range of light levels, and at adaptation luminances below 0.1 cd/m², 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/m². At 18° off-axis, vision is—according to our simple linear model—entirely dominated by rods at 0.06 cd/m², 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). 

At 42 cd/m², 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/m² 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/m² 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/m² 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/m², whereas the rod response becomes insignificant. 

It is noteworthy that in Figure 6 the increase in short-wavelength sensitivity with eccentricity is not confined to the parafoveal region where macular pigment is a recognized influence but continues progressively at eccentricities beyond 5°. We suggest in the Discussion section that this extrafoveal sensitivity profile, like the decreased blue sensitivity near the fovea, may be an expression of the distribution of the centrally concentrated short-wavelength “macular” pigment. 

The results of Figures 5 and 6 may be characterized to a first approximation by defining the luminance of equal rod and cone contribution for a particular eccentricity, which we call the “meso-mesopic” luminance. The meso-mesopic luminance M shown in Figure 7 in units of scotopic cd/m² is strongly dependent on eccentricity. It increases from 0.04 sc cd/m² at 2° eccentricity to 0.13 sc cd/m² at 5° and 0.44 sc cd/m² 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. 

Figure 7

 

The meso-mesopic luminance M, defined as the luminance of equal rod and cone contribution shown for several eccentricities of the stimulus (=radius of the annulus in degrees). The unit of M is scotopic cd/m².

Table 2

 

Modeled TVI slopes k of the rod system, the meso-mesopic luminance M in scotopic cd/m², and RMSE of the fit for stimuli radii between 2° and 18°. The model is based on the averaged relative photopic weights of the 8 observers as shown in Figure 5 (see 2 for details).

Radius of the annulus in degrees	k	M (in sc cd/m2)	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

Modeling the effects of changes in intensity

The differences in sensitivity between the CIE₁₉₆₄ 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. 

The variation in the relative cone weight with adaptation level was modeled with a sigmoid function (Equation 5) of the scotopic sensation luminance, 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 

$W_{\mathrm{P}}=1/(1+(M/S^{\prime }{)}^k).$

(5)

 

The scotopic adaptation luminance 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). 

Figure 8

 

Relative cone weights vs. scotopic adaptation luminance for stimulus radii between 1° and 18°. For radii of 5° and larger, the relative weights are based on the average photopic and scotopic sensation luminances.

Table 2 summarizes the values of k, M, and RMSE. 

Variation in stimulus color: Are photopic and scotopic luminance sufficient?

The question arises whether the two quantities, scotopic and photopic luminance, are sufficient to describe a mesopic luminance response, as most models assume (e.g., Palmer, 1966; Rea et al., 2004). This is by no means clear a priori, since four photoreceptor types (three cones and rods) combine in ways not well understood to determine mesopic luminance, which in principle may be a function of all four photoreceptor excitations rather than just the photopic and scotopic values of the stimulus. To investigate this point, the equal energy white field was replaced either by a purple or a green field of the same 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. 

Two subjects were tested at adaptation levels ranging from photopic to low mesopic. Figures 9a and 9b show the relative cone weights for the equal energy white and for the purple and green adaptation fields of equal photopic and scotopic luminance. For both observers, the cone weights for all conditions agree well. Though a statistical comparison of the weights shows, for some stimulus sizes, significant differences between the two conditions, the differences are minor and the results can be regarded as showing good agreement between the results for the differently colored fields. The model described earlier and in 2 predicts that the relative cone weight will be the same for differently colored backgrounds and stimuli that are matched in both photopic and scotopic values. This is in rough agreement with our observations. 

Figure 9

 

Comparison of the relative cone weights for an equal energy white background (solid lines), a purple background (dashed lines), and a green background (dotted lines) for several stimulus radii. (a) Observer 1 and (b) Observer 9 (all data for 2 Hz, 1 cpd, 4–5 repetitions). Observer 9 set the minimum motion nulls for the eew and purple adaptation fields only.

Separate manipulation of rod and cone adaptation level

Thus far, we have not distinguished between the adapting light levels for rods and for cones. The adapting stimuli to which the test modulations were applied either varied simply in overall intensity (Figure 5) or else varied in color in such a way as to keep the 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/m². 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/m², 0.054 photopic cd/m², and 0.29 photopic cd/m² (Figures 10 and 11). 

Figure 10

 

Relative cone weights as a function of S/P ratio of the background at a constant photopic luminance of 0.29 cd/m². Solid lines are for Observer 1, and dashed lines are for Observer 9.

Figure 11

 

Relative cone weights as a function of S/P ratio of the background. Solid lines are for Observer 1 who was tested at 0.027 cd/m², and dashed lines are for Observer 9 at a background luminance of 0.054 cd/m².

As expected at the field luminance of 24 photopic cd/m², 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. 

Figure 12

 

Rod weights vs. S/P ratio for the adaptation levels of 8 cd/m² (2 lower curves, obtained with radii of 5° (red) and 14° (black)) and 24 cd/m² (6 upper curves for the radii of 2° (blue), 5° (red), and 14° (black)) for two observers. Solid lines are for Observer 1, and dashed lines are for Observer 9.

In the model of 2, the variation in cone weight with 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. 

Table 3

 

The best-fitting equivalent TVI exponent k for mesopic adaptation conditions at a constant photopic luminance and varying scotopic luminance. See 2 for detailed explanations on their calculation.

P adapt (cd/m2)	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

Surprisingly, the 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). 

Change of red–blue isoluminance ratio with temporal frequency

For all data shown so far, the temporal frequency of the grating was fixed at 2 Hz. This frequency is low enough to allow the rods to participate in motion perception and to avoid noticeable phase disturbances between the rod and the cone signals (MacLeod & Stockman, 1987). These can lead to complete cancellation at around 7.5 Hz and are due to the sluggishness of the rod system compared to the cone system (MacLeod, 1972). Thus, flicker photometry fails to create good nulls unless the stimulus phase is appropriately adjusted. A key advantage of minimum motion photometry over flicker photometry is the ability to set motion nulls at low temporal frequencies where phase differences are small. 

Figure 13 shows that variation of temporal frequency between 0.5 Hz and 5 Hz had little or no effect on minimum motion isoluminance ratios for a mesopic luminance level of 0.09 cd/m². 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. 

Figure 13

 

Relative cone weights for an adaptation luminance of 0.09 cd/m² as a function of temporal frequency. All data points are averages from 5 settings by Observer 1. The stimulus spatial frequency was constant at 1 cpd.

Change of red–blue isoluminance ratio with spatial frequency

The spatial frequency of the stimulus was varied between 0.5 cpd and 4 cpd (8 cpd for photopic conditions; spatial frequencies above 4 cpd could not be resolved at the mesopic luminance levels tested). Figures 14 and 15 show for observers 1 and 9 that for all tested mesopic adaptation levels a distinct increase of the relative cone weights occurs at higher spatial frequencies. Even at 0.01 cd/m² 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/m² 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). 

Figure 14

 

Relative cone weights of Observer 9 as a function of spatial frequency. The colors of the curves correspond to the radius: 2° (blue), 5° (red), 14° (black). The adaptation luminances are 42 cd/m² (dashed lines, 2 topmost curves), 0.09 cd/m² (solid lines), and 0.01 cd/m² (dotted line, bottommost curve). Each data point is the average of 5 measurements from Observer 9. The temporal frequency was constant at 2 Hz.

Figure 15

 

Relative cone weights for an adaptation luminance of 0.09 cd/m² as a function of spatial frequency. All data points are averages over 5 measurements from Observer 1. The temporal frequency was constant at 2 Hz.

Discussion

The minimum motion technique proves capable of providing precise and reproducible estimates of mesopic luminance. As we discuss below, the results can be very simply and accurately modeled by representing mesopic luminance as an adaptation-dependent linear combination of scotopic and photopic contributions, in which the weights are set mainly by the scotopic luminance of the adapting field. 

Our results show a progressive transition from cone vision to rod vision as luminance increases, with the relative photopic weight increasing from 0 to 1. However, the shift from cone to rod vision during decreasing adaptation levels does not occur uniformly over the visual field. For off-axis vision, the rods take over at relatively bright light levels, whereas near foveal vision (±2°) is dominated by the cones down to low mesopic light levels. This effect is expected, since it is determined by the different spectral sensitivity of the receptors and their distinct distributions over the retina (Curcio et al., 1991). The influence of eccentricity is highest at mid-mesopic light levels, where vision is dominated by the cones in foveal and near-foveal areas and by the rods at 18° off-axis. Accordingly, the “meso-mesopic luminance”, defined as the adaptation luminance for equal rod and cone contribution to luminance, increases with eccentricity (see Figure 7). But, the entire mesopic transition, expressed by the change in relative cone weight with log luminance, has nearly the same shape for all eccentricities apart from a translation to accommodate the change in the meso-mesopic luminance (Figures 5 and 8). 

The effect of eccentricity at various adaptation conditions with a minimum motion stimulus was examined earlier by Anstis (2002). He found that sensitivity for short-wavelength light increases linearly with eccentricity. The slope of this gradient increased by a factor of 2 to 4 with each log₁₀ unit reduction of luminance. The present results are somewhat different: At photopic light levels, the blue sensitivity measured as the isoluminance ratio ΔP _R/ΔP _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. 

The change of sensitivity with eccentricity for pure rod and cone vision

We were surprised to find systematic deviations from the standard photopic and scotopic sensitivities at both the high and low extremes of the luminance range. At photopic levels, the isoluminance ratios between red and blue agree with the 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/m², decreasing to 0.008 cd/m² 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. 

One possible candidate explanation for the decrease in the relative cone weight for peripheral stimuli is that it reflects residual rod activity at 42 cd/m². Rods begin to saturate around 2.0 log₁₀ scotopic trolands (sc td; ≈3 cd/m²) but are not completely saturated until 120–300 cd/m² (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. 

Another possible reason for the change in isoluminance ratio at scotopic and photopic adaptation levels might lie in some blue-absorbing macular pigment still present at a radius of 5°. On the basis of the color matching functions of Stiles and Burch (1955), Stockman, MacLeod, and Johnson (1993) estimated a macular pigment peak density of 0.15 for a full 10° field. The estimate of a standard deviation of 0.12 in peak macular pigment density among observers for a 10° field by Webster and MacLeod (1988) could suggest an even greater mean density than this. So it is not surprising that the luminosity functions continue to shift at large eccentricities if macular pigment is still significant at 5°. However, to compensate for the shift of spectral sensitivity at 42 cd/m² 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/m², 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.) 

There is, however, almost no clear evidence about the variation in macular pigment density at large eccentricities, so the large gradients required to explain our results on the basis of macular pigment alone cannot be excluded. Both psychophysical and objective estimates of the profile have generally assessed the spatial profile relative to a largest measured eccentricity less than 8°. In place of actual measurement of the outer reaches of the profile, it has become accepted practice to assume that the density at the largest measured eccentricity is negligible. This practice has been encouraged by the adoption of a second poorly supported assumption that MPOD varies exponentially with eccentricity. The actual results of the investigations where these assumptions are invoked (Barbur et al., 2010; Berendschot & van Norren, 2006; Hammond, Wooten, & Snodderly, 1997; Moreland, 2004; Nolan, Stringham, Beatty, & Snodderly, 2008; Robson et al., 2005; Robson & Parry, 2008; Vienot, 2001) suggest a linear rather than an exponential decline of MPOD with eccentricity in the periphery, with a gradient sufficient to generate a difference between 0.05 and 0.10 between eccentricities of 4° and 8°. Extrapolation on that basis leaves open the possibility that the MPOD is of the order of 0.1 or 0.2 at an eccentricity of 10° and may be considerably more for observers whose MPOD exceeds the average. Thus, macular pigmentation could conceivably be the entire explanation for the eccentricity dependence of our scotopic and photopic sensitivities. 

However, there are other possible causes of an increase in peripheral blue sensitivity relative to red. One possibility is a reduction in optical photopigment density of L and M cones at larger eccentricities. Stockman et al. (1993) estimated a decrease of photopigment density of 0.1 between the 2° observer of Stiles and Burch (1955) and the 1964 CIE 10° observer. However, there is no clear evidence in the literature showing a systematic change in relative sensitivity to red and blue for eccentricities beyond 5° off-axis (Knau, Jägle, & Sharpe, 2001; Stabell & Stabell, 1980). Alternatively, a change of L and M cone weights with eccentricity could explain the eccentricity dependence of photopic sensitivity, but no such change was evident in the studies by Knau et al. (2001) and Nerger and Cicerone (1992). A further consideration is the slight nonuniformity of the CRT monitor (the relative luminance of the red and blue phosphors was not quite uniform), but calibration checks showed that this was not enough to contribute substantially to the effect. 

The rationale for the scotopic exponent model of mesopic luminance

The decrease of the relative cone weights with decreasing adaptation luminance was modeled as a simple sigmoid function (Equation 5) with two parameters: the eccentricity-dependent meso-mesopic luminance 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. 

The effects of changing background color complicate the picture slightly. When comparing relative cone weights under adaptation to backgrounds of the same photopic luminance but of different color and different scotopic luminance, background colors of high 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. 

The rod sensitivity exponents exceeding 1 with changes in background color have a possible parallel in detection threshold observations, where strong cone stimulation by reddish adapting fields reduces rod sensitivity, as if by a winner-take-all mutual inhibition between rods and cones (Makous & Boothe, 1974; Sharpe, Fach et al., 1989). A model amended along these lines could accommodate the data of Figures 10–12 as follows. In those figures, an increase in the 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. 

Photopic luminance is, by convention, additive and linearly related to the cone excitations. However, even photopic luminance involves adaptation and nonlinearity, with adaptation-dependent weights combining the contributions of the L and M cones; consequently, flicker photometry is strictly only incrementally additive, for small perturbations around a fixed state of adaptation (Eisner & Macleod, 1981). Our mesopic luminance model likewise implies 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. 

The model of Equation 5 makes the relative cone weight a function of scotopic but not photopic adaptation. This differs from other proposals (e.g., Palmer, 1968; Rea et al., 2004). In Palmer's (1968) model, for example, the effectiveness of scotopic input is independent of rod adaptation but dependent on cone adaptation. Our simple model has a theoretical rationale in the behavior of rod and cone TVI curves where the curve for cones has zero or shallow slope at mesopic levels, and our experiment supports it by the agreement of our exponent k (that determines the width of the mesopic log intensity range) with the corresponding value from the TVI curve slope. 

The just discussed effects of changes in background color (Figures 10–12) do restrict the generality of Equation 5, owing to the greater value of 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. 

Finally, Figure 8 makes it clear that the effects of variation in eccentricity can be fairly well accommodated by variation in the meso-mesopic luminance M alone; this horizontally translates the fixed curve shape from Equation 5 along the log intensity axis. 

Comparison of a neutral gray and a purple field of equal S/P ratio

To test whether scotopic and photopic luminance alone are sufficient to describe mesopic luminance, purple and greenish fields of different color but equal 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). 

Change of red–blue isoluminance ratio with spatial and temporal frequency

Under mesopic adaptation conditions, the different spatial and temporal response properties of the rod and cone system jointly determine the total response and the results would, therefore, be expected to depend on the spatiotemporal structure of the stimulus. The spatial resolution of the cone system is known to be higher than that of the rod system when foveal cones are compared with extrafoveal rods or when cone vision at high light levels is compared with rod vision at scotopic levels. But, there is little evidence about the relation between rod and cone spatial resolution when conditions of stimulation and retinal location are comparable. Such a comparison may be made by considering the behavior of the rod and cone weights in the present experiment as a function of spatial frequency. The results show that rod and cone vision do differ in spatial resolution, even when tested with similar stimuli at the same retinal position and luminance. At intermediate mesopic levels, a clear increase in cone weight is evident as spatial frequency is increased (Figures 14 and 15). At 10° and 0.09 cd/m², 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. 

The basis for this inferiority of rod vision is not yet clear. The rod photoreceptors are small enough and densely enough packed to allow spatial resolution even higher than the parafoveal cones, and there is no resolution-degrading electrical crosstalk between neighboring rods as there is for cones (Field & Chichilnisky, 2007). Postreceptorally, the A II amacrine interneuron links rods to most or all retinal ganglion cell types via cone bipolars. The physiological effectiveness of those connections has not been fully quantified, so it is possible to suppose that rod signals relayed by the A II amacrine are excluded from the most highly resolving ganglion cell arrays (Field et al., 2009). But, rod signals also have direct access to cone pathways through electrical connections with the cones. We might expect this “gap junction” signal route to allow high spatial resolution for rod-isolating stimuli (e.g., Sharpe, Stockman, & MacLeod, 1989). This remains a puzzle for future investigation. 

Measuring the temporal response properties under mesopic conditions introduces the issue of phase lags within the rod system and between the rod and cone systems. To avoid enhancement and attenuation, we used low temporal frequencies where an unambiguous motion null could be found. We found no influence of temporal frequency at 5 Hz and below. This agrees with minimal or zero effects in studies by Cavanagh and Anstis (1991) and Cavanagh et al. (1987) under photopic conditions. The limited influence of temporal frequency on the rod weight in our experiments is surprising given the evidence of more limited sensitivity of the rod system to high temporal frequencies (Conner, 1982; Kelly, 1961). 

Individual variation in photopic isoluminance

Taking a closer look at the isoluminance ratios of the individual observers at 42 cd/m² 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. 

Figure 16

 

Isoluminance ratios at 42 cd/m² for radii of 10°, 14°, and 18°. For each annulus radius, the isoluminance ratios of all subjects are shown in identical order. The error bars are the single ± standard deviation. The two colors represent the two groups that might reflect different variants of L cone pigments.

Influence of pupil size

In all experiments described here, the natural pupil was used. Pupil size measurements at the two highest adaptation levels of 42 cd/m² and 2 cd/m² 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/m² and 0.002 cd/m²—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%. 

Conclusion

This paper examines mesopic luminosity under a variety of conditions as determined by motion nulling with a first-order motion stimulus. Motion nulling is a precise and unambiguous method to assess achromatic luminance information under adaptation conditions ranging from photopic to scotopic. By using temporal frequencies of 2 Hz or below, considerable phase lags between rods and cones and therefore cancellations of receptor signals can be avoided. Thus, this method does not inherit the drawbacks of flicker photometry, which was used to define the scotopic and photopic standard observers but is problematic in mesopic vision. 

The complex nature of mesopic vision has made mesopic luminosity a challenging area, especially for practical applications. Earlier approaches to determine a mesopic equivalent of luminance have emphasized on tasks like detection of colored targets (Goodman et al., 2007) and heterochromatic brightness matching (e.g., Kokoschka, 1980; Sagawa & Takeichi, 1992), which are known to involve achromatic and chromatic information of the visual system, hence do not reflect a measure of achromatic luminance. Here, we offer a method to determine luminance as defined in flicker photometry for all adaptation conditions under a variety of retinal positions. 

Determining luminance under dim conditions has practical importance since luminance photometry does not and cannot account for the complexities of mesopic perception. However, for many safety-related applications like emergency lighting and vehicle and street lighting, a perception-based determination of photometric measures is highly desired. 

Yet, a unified system for mesopic luminosity remains a task for the future. 

Appendix A

The calculation of photopic and scotopic sensation luminance

The photopic sensation luminance was defined by weighting the luminance contribution of the blue and red phosphors to provide for isoluminance at the intensities required in a minimum motion null at 42 cd/m². 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: 

$P^{\prime }=r{\omega }_{\mathrm{R}\mathrm{,}\mathrm{p}\mathrm{h}\mathrm{o}\mathrm{t}}{P}_{\mathrm{R}}+g{P}_{\mathrm{G}}+b{\omega }_{\mathrm{B}\mathrm{,}\mathrm{p}\mathrm{h}\mathrm{o}\mathrm{t}}{P}_{\mathrm{B}}.$

(A1)

Here, 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/m²) 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. 

Figure A1 (2) shows the relative cone weights based on photopic and scotopic luminance and photopic and scotopic sensation luminance for one stimulus size. 

Figure A1

 

Relative cone weight vs. scotopic adaptation luminance for an eccentricity of 18°. The weights are based on the CIE photopic and scotopic luminosity functions (blue curve) and the sensation luminances P′ and S′ (red curve).

Appendix B

Modeling the effects of changes in intensity and color

The variation in the relative cone weight with adaptation luminance based on scotopic and photopic sensation luminances, S′ and P′, can be described with a sigmoid function. 

The following assumptions are made to simplify the modeling of the data: 

1.  
   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 

   ${\Delta }^{*}S\sim S^k.$

   (B1)
2.  
   The cone incremental threshold is treated as constant at adaptation conditions below 40 td (∼2.5 cd/m²). 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′.
3.  
   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: 

   $W_{\mathrm{S}}^{\prime }\sim S^{-k}.$

   (B2)

On the assumption that the independent cone weight W _P′ is constant and that rods and cones do not interact, this relation leads to 

$W_{\mathrm{P}}=1/(1+(M/S^{\prime }{)}^k),$

(B3)

where 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/m². The scotopic standard observer adapting luminance S has here been replaced by its sensation luminance counterpart S′. 

Figure A1 shows the fitted curve for iteratively optimized parameters M and k for an 18° annulus. 

A bluish adapting stimulus with a high 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. 

Acknowledgments

This work was supported by NIH Grant EY01711 to DIAM and by a grant to SR from the Deutscher Akademischer Austausch Dienst. 

Commercial relationships: none. 

Corresponding author: Sabine Raphael. 

Email: [e-mail]. 

Address: Gleueler Str. 50, Cologne 50931, Germany.