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Article  |   February 2025
Psychophysically measuring the efficiency of rods
Author Affiliations
  • Geneviève Rodrigue
    School of Optometry, Université de Montréal, Montréal, QC, Canada
    [email protected]
  • Laurine Paris
    School of Optometry, Université de Montréal, Montréal, QC, Canada
    [email protected]
  • Judith Renaud
    School of Optometry, Université de Montréal, Montréal, QC, Canada
    [email protected]
  • Rémy Allard
    School of Optometry, Université de Montréal, Montréal, QC, Canada
    [email protected]
Journal of Vision February 2025, Vol.25, 1. doi:https://doi.org/10.1167/jov.25.2.1
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      Geneviève Rodrigue, Laurine Paris, Judith Renaud, Rémy Allard; Psychophysically measuring the efficiency of rods. Journal of Vision 2025;25(2):1. https://doi.org/10.1167/jov.25.2.1.

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Abstract

Recent studies suggest that the efficiency of cones to detect photons can be evaluated by measuring the equivalent input noise (EIN; derived from contrast thresholds measured in the presence and absence of visual noise) under specific conditions in which the contrast threshold is limited by the variability in the number of photons detected by photoreceptors (i.e., photon noise). These conditions can be identified based on the known properties of photon noise: spatially and temporally white and inversely proportional to the luminance intensity. The present study aims to adapt this psychophysical paradigm to evaluate the efficiency of rods to detect photons. A motion direction discrimination task was used to evaluate the EIN over a wide range of luminance intensities for various spatial and temporal frequencies when the display was blue or red (to which rods have little sensitivity). The target was either a Gabor patch presented at 20 degrees of eccentricity (first experiment) or a rotating sine-wave annulus with a radius of 10 degrees of eccentricity (second experiment). In both experiments, the EIN was found to be inversely proportional to luminance intensity over a limited range of luminance intensities for both display colors. At these luminance intensities, the EIN was roughly independent of the spatial and temporal frequencies, matching the properties of photon noise. Furthermore, under these conditions, contrast thresholds were lower (i.e., better) when the display was blue rather than red, which suggests that vision was mediated by rods when the display was blue. We conclude that the efficiency of rods to detect photons can be evaluated by measuring contrast thresholds in the presence and absence of visual noise over a limited range of luminance intensities with a blue display.

Introduction
The function of photoreceptors is to detect photons and convert them into neural signals. The two types of photoreceptors, cones and rods, enable photopic (i.e., daylight) and scotopic (i.e., dim light) vision, respectively. Interestingly, the efficiency of cones to detect photons can be evaluated psychophysically (Braham Chaouche, Rezaei, Silvestre, Arleo, & Allard, 2021; Silvestre, Arleo, & Allard, 2018), which has been used to evaluate the impact of healthy aging on the efficiency of cones (Braham Chaouche, Silvestre, Trognon, Arleo, & Allard, 2020; Silvestre, Arleo, & Allard, 2019). This psychophysical paradigm was found to be more sensitive to the efficiency of cones to detect photons than standard clinical tests such as visual acuity and Pelli–Robson contrast sensitivity (Rezaei, Chamaa, Rodrigue, Renaud, & Allard, 2021). This suggests that this functional measure could be more sensitive to mild photoreceptors deficiencies than current standard clinical tests. The aim of the current study was to adapt this psychophysical paradigm to evaluate the efficiency of rods to detect photons, which could be useful, for example, to compare the impact of retinal diseases on the efficiency of two types of photoreceptors. 
Photon noise
The efficiency of photoreceptors can be evaluated psychophysically by evaluating the photon noise (Braham Chaouche et al., 2021), which represents the variability in the number of photons detected by photoreceptors in the presence of a constant light (Pelli, 1990). Indeed, because the detection of a photon by photoreceptors is stochastic, the number of photons detected by a photoreceptor varies in time even in the presence of a constant light (Braham Chaouche et al., 2021; Silvestre et al., 2018), and this variability in the response of photoreceptors is a source of internal noise that may affect contrast sensitivity. Knowing the variance in the number of photons detected is sufficient to estimate the number of photons detected by photoreceptors because the number of photons detected follows a Poisson distribution (Pelli, 1990), so the variance in the number of photons detected (i.e., the photon noise) is equal to the average number of photons detected by the photoreceptors. As a result, the variability in the number of photons detected is a source of internal noise that may affect contrast sensitivity and is directly related to the number of photons detected. 
The impact of the internal noise affecting contrast sensitivity can be evaluated using an external noise paradigm by measuring contrast sensitivity in different levels of external noise added to the display (Pelli, 1981; Pelli, 1990; Pelli & Farell, 1999). If the external noise added to the display is weaker than the internal noise limiting contrast sensitivity, then its impact on contrast sensitivity will be negligible. On the other hand, if the external noise is stronger than the internal noise, then it will affect contrast sensitivity. The amount of external noise added to the display that has the same impact as the internal noise is referred to as the equivalent input noise (EIN). The EIN can be estimated based on two contrast thresholds (Braham Chaouche et al., 2021; Pelli & Farell, 1999), typically with and without visual noise added to the display, and provides an estimate of the impact of the internal noise affecting contrast sensitivity. 
However, photon noise is not the only source of internal noise that can limit contrast sensitivity, as there is internal noise all along the visual pathway. Nevertheless, only the stronger source of noise considerably affects contrast sensitivity, as the weaker sources of noise have a negligible impact on sensitivity (Pelli, 1981; Silvestre et al., 2018), and, under some conditions, the dominant source of noise limiting contrast sensitivity is the photon noise (Braham Chaouche et al., 2020; Braham Chaouche et al., 2021; Silvestre et al., 2018). The main aim of the current study was to seek the conditions under which the photon noise originating from rods is the dominating source of noise and can thereby be evaluated using an external noise paradigm. 
Different sources of internal noise have different properties, so the source of internal noise limiting contrast sensitivity can be inferred based on the measured properties of the internal noise limiting contrast sensitivity (Silvestre et al., 2018). When photon noise is the dominating source of noise, contrast sensitivity should depend on the properties of the photon noise. Most importantly, because the number of photons detected follows a Poisson distribution (Pelli, 1990), the variance in the number of photons detected is proportional to the luminance intensity, which predicts the de Vries–Rose law (de Vries, 1943; Rose, 1948); that is, the EIN should be inversely proportional to the luminance intensity (Braham Chaouche et al., 2021; Silvestre et al., 2018). Furthermore, because the number of photons detected by a photoreceptor is independent of the number of photons detected by other photoreceptors, photon noise is expected to be spatially white (Pelli, 1990), and, because the stochastic detection of a photon by a photoreceptor has a negligible impact on the probability of other photons to be detected by the same photoreceptor, photon noise is also expected to be temporally white (Graham & Hood, 1992; Pelli, 1990). In sum, if photon noise is limiting contrast sensitivity, we should expect the EIN to be inversely proportional to the luminance intensity and spatiotemporally white. 
Testing rods and cones separately
An important goal of the current study was to assess independently the efficiency of rods and cones to detect photons; this can be achieved by evaluating the photon noise under scotopic and photopic vision, respectively. Because rods are absent in the foveola (Curcio, Sloan, Kalina, & Hendrickson, 1990), an eccentric stimulus is required to assess the efficiency of rods. Outside of the foveola, rod density increases with eccentricity and peaks in the perifoveal zone around 18 degrees of eccentricity, whereas cone density is maximal at the center of the foveola and decreases rapidly toward the periphery (Curcio & Owsley, 2019; Curcio et al., 1990). Another important difference between rods and cones is that they have different spectral sensitivities, as rods have little sensitivity to long wavelengths. Thus, the perception of long wavelengths (i.e., a red stimulus) is more likely to be enabled by cones, whereas the perception of short wavelengths (i.e., a blue stimulus) is expected to be enabled by rods at low luminance intensities and cones at high luminance intensities. 
In sum, the aim of the current study was to adapt the psychophysical paradigm to assess the efficiency of rods to detect photons by finding the conditions under which photon noise properties can be observed—that is, EIN that is inversely proportional to luminance intensity (i.e., de Vries–Rose law) and spatiotemporally white. Furthermore, if rods considerably contribute to contrast sensitivity when using a blue display, then contrast sensitivity is expected to be poorer when using a red display at the same photopic luminance intensity (i.e., with a similar number of photons detected by cones for red and blue displays). On the other hand, if rods do not considerably contribute to contrast sensitivity when using a blue display, then contrast sensitivity is expected to be similar using blue and red displays at the same photopic luminance intensities. 
Experiment 1. Localized peripheral stimulus
To evaluate the efficiency of rods, contrast sensitivity must be measured under conditions in which vision is mediated by rods. In the first experiment, the visual target was a small stimulus presented in perifoveal retina, where the density of rods is much higher than the density of cones. 
Method
Participants
Four healthy young adults between the ages of 18 and 35 years participated in the first experiment. No participant had a known visual or systemic pathology, and their best-corrected visual acuity was 20/25 or better in their dominant eye, which were the inclusion criteria to participate in the study. None of the participants was taking medication that could impair their vision. Exclusion criteria included a history of eye diseases, systemic diseases affecting vision, and the use of medications that may affect vision. This study adhered to the tenets of the Declaration of Helsinki, and ethical approval was obtained by Université de Montréal's Comité d’éthique de la recherche Clinique. 
Apparatus
The computer display was a 22.5-inch VIEWPixx LCD screen (VPixx Technologies, Saint-Bruno, QC, Canada) designed specifically for psychophysics and was the only source of light in the room. The spatial resolution of the display was 1920 × 1080 pixels, and its refresh rate was 120 Hz. At the 50-cm viewing distance, the spatial resolution was 32 pixels per degree of visual angle. A CS-1000 spectroradiometer (Konica Minolta, Tokyo, Japan) was used to linearize the output intensity of each color gun. The 8-bit display was perceptually equivalent to an analog display having a continuous luminance resolution by using the noisy-bit method (Allard & Faubert, 2008) implemented independently for each color gun. 
Stimuli
The target was a sine-wave grating with a spatial frequency of 0.5 cycle per degree (cpd) presented in the middle of the display. The participant gazed monocularly at a white fixation point so that the target was presented at 20° in the nasal visual field, thus stimulating the temporal retina (Figure 1). The fixation point was white to optimize its visibility under the darkest conditions. The sine wave drifted at a temporal frequency of 1, 2, 4, or 8 Hz, depending on the block condition. The display color was either blue or red (i.e., only the blue or red display color was used, respectively; see their spectral densities in Figure 2). The target was presented for 250 ms plus on and off half-cosine ramps of 125 ms. The spatial window of the target was a circle of 2° plus a half-cosine soft edge of 0.5°. The target was presented in either the absence or presence of visual noise added to the display. The noise added to the display was truncated filtered noise (Jules Étienne, Arleo, & Allard, 2017) with a low-pass filter with a cutoff frequency of 1 cpd, temporally white (refreshed at 60 Hz), truncated at 1 SD (i.e., applying a maximum and minimum to noise samples after applying the filter), and with its contrast set to 50%, resulting in a noise energy of 657 µs deg2. To avoid triggering a change in processing strategy, which would violate an underlying assumption of the external noise paradigm (Allard & Cavanagh, 2011; Allard & Faubert, 2014a; Allard & Faubert, 2014b; Allard, Renaud, Molinatti, & Faubert, 2013), the dynamic noise was continuously presented over the entire display. The luminance intensities of the red and blue displays were set to 12.4 and 5 photopic cd/m2, which corresponded to 0.8 and 21 scotopic cd/m2, respectively. The luminance intensity was varied using neutral-density filters. 
Figure 1.
 
Stimuli examples presented on the blue display without external noise (top) and the red display with external noise (bottom) during the first experiment. The target was a sine-wave grating, and the participant was asked to gaze at the white fixation point presented at 20° to the left or right of the target, depending on whether the participant viewed the target with his left or right eye, respectively.
Figure 1.
 
Stimuli examples presented on the blue display without external noise (top) and the red display with external noise (bottom) during the first experiment. The target was a sine-wave grating, and the participant was asked to gaze at the white fixation point presented at 20° to the left or right of the target, depending on whether the participant viewed the target with his left or right eye, respectively.
Figure 2.
 
Normalized spectral energy distribution of the blue and red color gun emitted by the display.
Figure 2.
 
Normalized spectral energy distribution of the blue and red color gun emitted by the display.
The participant was asked to report the motion direction of a target (up or down) by pressing one of two keys. A feedback sound was provided to indicate the correctness of the answer. The contrast of the target was controlled using a three-down, one-up staircase procedure (Levitt, 1971) interrupted after 14 inversions. The threshold was estimated as the geometric mean of the last 10 inversions. 
Procedure
Before the psychophysical experiment began, corrected visual acuity of each eye was measured using an Early Treatment Diabetic Retinopathy Study (ETDRS) chart at 4 meters in a clinical room at the Clinique Universitaire de la Vision at the Université de Montréal. If the participant was wearing optical correction, the prescription was analyzed with a manual lensometer and put in the trial frame for the testing. 
For the psychophysical experiment, two experimental sessions were planned on separate days: one with the red display and one with the blue display. The participant wore a trial frame with the non-dominant eye occluded with an absolute black occluder, and their prescription with neutral-density filters (depending on the luminance intensity condition) was placed in front of the dominant eye. Before the beginning of the psychophysical experiment, the participant was first adapted to the lowest luminance intensity for a period of at least 30 minutes using the darkest filter added to the trial lenses. After the adaptation phase, the pupil size was measured using an electronic pupillometer (VIP-300; NeurOptics, Irvine, CA). To avoid disrupting adaptation, the participant closed their eyes during setup of the pupillometer. While the participant's eyes were closed, the trial frame was removed. The electronic pupillometer covered the dominant eye (i.e., the viewing eye during the experiment) without emitting any light into it, and neutral-density filters were held in front of the non-dominant eye. The participant then opened their eyes and gazed at the display through the neutral-density filters with the non-dominant eye. When the pupil size was stabilized, which was determined using the image of the pupil through the pupillometer screen, after approximatively 15 seconds the pupil size was estimated based on the average of three consecutive measures. Afterward, the participant closed their eyes once again, and the trial frame was reinstalled with their prescription and the neutral-density filters in front of their dominant eye and the black occluder in front of their non-dominant eye. 
Each experimental session started with the darkest condition, and luminance intensity was gradually increased using different neutral-density filters to avoid the necessity of re-adapting to lower luminance intensities. During the transition between two luminance intensities, the participant closed their eyes during setup of the pupillometer to record the pupil size (as described above) and when reinstalling the trial frame with less neutral-density filters to increase luminance intensity. At the beginning of each luminance intensity block, which was always brighter than the previous one, the participant was adapted to the new luminance intensity for about 1 minute, and then pupil size was measured following the same procedure as above. 
For each luminance intensity block, contrast threshold was measured at each temporal frequency in a random order. Finally, contrast thresholds were measured in the presence of visual noise added to the display at the highest luminance intensity at each temporal frequency. Because the contrast threshold in noise is known to vary little with the luminance intensity (Nagaraja, 1964; Pelli, 1981), it was measured at only one luminance intensity and assumed to be independent of the luminance intensity (e.g., Silvestre et al., 2018; Silvestre et al., 2019). Nonetheless, we did address this assumption later in the current study (see Experiment 2b), where contrast thresholds in high visual noise were evaluated as a function of luminance intensity, which validated this initial assumption. 
Estimating the equivalent input noise
Based on the linear-amplifier model (Pelli, 1981; Pelli, 1990), the contrast threshold (c) as a function of the external noise energy (Next) can be modeled using the following function:  
\begin{eqnarray} c\left( {{{N}_{{\rm{ext}}}}} \right) = \sqrt {\frac{{{{N}_{{\rm{eq}}}} + {{N}_{{\rm{ext}}}}}}{k}}\quad \end{eqnarray}
(1)
where Neq represents the EIN (i.e., the external noise energy that has the same impact as the internal noise), and k represents the calculation efficiency (i.e., directly related to the signal-to-noise ratio required to detect the signal). 
With high external noise, where c(Next) >> c(0), the impact of the internal noise is negligible (Neq ≈ 0), and Equation 1 can be used to derive the calculation efficiency (k) from the contrast threshold in high noise (Pelli & Farell, 1999):  
\begin{eqnarray} k = \frac{{{{N}_{{\rm{ext}}}}}}{{{{c}^2}\left( {{{N}_{{\rm{ext}}}}} \right)}}\quad \end{eqnarray}
(2)
 
Given the calculation efficiency, k, and the contrast threshold in absence of noise, c(0), Equation 1 can be used to derive the EIN:  
\begin{eqnarray} {{N}_{{\rm{eq}}}} = {{c}^2}\left( 0 \right)k\quad \end{eqnarray}
(3)
 
Given that the impact of the photon noise is inversely proportional to the luminance intensity (L), then when the dominant source of noise is the photon noise the EIN can be modeled with the following equation (Silvestre et al., 2018):  
\begin{eqnarray} {N}_{\rm eq\, fit}( L ) = \frac{{{{N}_{{\rm{photon}}}}}}{L}\quad \end{eqnarray}
(4)
 
Results and discussion
Figure 3 represents the contrast thresholds in the absence of noise as a function of the photopic luminance intensity for different temporal frequencies when the display was red and blue. At low luminance intensities, contrast thresholds were lower when the display was blue than when it was red, suggesting that vision was mediated by rods when the display was blue, as expected under such low luminance intensities (<∼5 photopic Td, which is approximatively 0.2 photopic cd/m2 and corresponds to mesopic vision—that is, when both rods and cones operate). 
Figure 3.
 
Contrast thresholds in the absence of noise as a function of photopic luminance intensity for each participant (different graphs) for different temporal frequencies (different symbols) when the display was red or blue (red and blue lines, respectively).
Figure 3.
 
Contrast thresholds in the absence of noise as a function of photopic luminance intensity for each participant (different graphs) for different temporal frequencies (different symbols) when the display was red or blue (red and blue lines, respectively).
Figure 4 represents the contrast thresholds in noise as a function of the temporal frequency for red and blue displays. Similar contrast thresholds were observed when the display was blue or red suggesting that the calculation efficiency, which depends on the contrast threshold in high noise (Equation 2), was independent of the display color. Thus, the calculation efficiency (k) was assumed to be the same for the two display colors and was derived using Equation 2 for each temporal frequency from the geometric mean of the contrast thresholds in noise for the red and blue displays. 
Figure 4.
 
Geometric mean contrast thresholds in noise measured when the display was red or blue (respective lines) as a function of temporal frequency. The error bars represent the standard error of the mean computed in log units.
Figure 4.
 
Geometric mean contrast thresholds in noise measured when the display was red or blue (respective lines) as a function of temporal frequency. The error bars represent the standard error of the mean computed in log units.
The EIN was derived for different luminance intensities and temporal frequencies and for the two display colors based on the calculation efficiency (k) estimated at each temporal frequency and the contrast thresholds in the absence of noise, c(0), at different temporal frequencies and different luminance intensities and for different display colors (Equation 3Figure 5). Whereas the results show some variability, the EIN measured was found to roughly follow the de Vries–Rose law (i.e., log–log slope of –1) over all luminance intensities tested on the red display and at low luminance intensities on the blue display (Figure 5). 
Figure 5.
 
EIN (derived from the data presented in Figures 3 and 4) as a function of photopic luminance intensity for each participant (different graphs). The thick lines illustrate the de Vries–Rose law (i.e., log–log slopes of –1), representing the impact of photon noise (see Equation 4) when the display was red or blue (respective colors) for each participant.
Figure 5.
 
EIN (derived from the data presented in Figures 3 and 4) as a function of photopic luminance intensity for each participant (different graphs). The thick lines illustrate the de Vries–Rose law (i.e., log–log slopes of –1), representing the impact of photon noise (see Equation 4) when the display was red or blue (respective colors) for each participant.
For each testing condition (two display colors and four temporal frequencies) and each participant, the photon noise (Nphoton) was estimated over the luminance intensities at which the EIN roughly followed the de Vries–Rose law (i.e., a log–log slope of –1; see fits in Figure 5). As can be seen in Figure 5, the de Vries–Rose law was generally observed for each condition (i.e., temporal frequency and color display) over a range of about three data points, so we arbitrarily chose to fit the photon noise to the second lowest photon noise (i.e., only one data point below the fitted line with a slope of –1). This arbitrary fit gave a reasonably good fit of the EIN over the luminance intensities at which the de Vries–Rose law was observed. 
Figure 6 presents the fitted photon noise at 1 Td (i.e., Nphoton in Equation 4) averaged across participants (geometric mean) as a function of temporal frequency for the red and blue displays. A repeated-measures 2 × 4 analysis of variance (ANOVA) on the photon noise in log units showed a significant large effect of color, F(1,3) = 26.589, p = 0.014, ω2 = 0.501, and a significant moderate effect of temporal frequencies, F(3,9) = 6.951, p = 0.010, ω2 = 0.058. No significant interaction was found between temporal frequencies and color, F(3,9) = 2.264, p = 0.150, ω2 = 0.007. The effect of color shows that the photon noise was lower when the display was blue than when it was red, which suggests that more photons were detected by rods than cones, as expected. Post hoc analyses using Bonferroni–Holm correction showed significant differences among 1 to 8 Hz, 2 to 8 Hz, and 4 to 8 Hz (p = 0.020, p = 0.019, and p = 0.045, respectively), suggesting that the photon noise was higher at 8 Hz than at lower temporal frequencies. However, photon noise may not have been the limiting noise source at 8 Hz when the display was blue, as the EIN did not clearly follow the de Vries–Rose law for some participants, and fewer data points could be measured at this frequency. Thus, photon noise was clearly found to be different with red and blue displays, but no clear evidence was found suggesting that the photon noise considerably varied with temporal frequency. 
Figure 6.
 
Geometric mean photon noise measures when the display was red or blue (respective lines) as a function of temporal frequency. The error bars represent the standard error of the mean computed in log units.
Figure 6.
 
Geometric mean photon noise measures when the display was red or blue (respective lines) as a function of temporal frequency. The error bars represent the standard error of the mean computed in log units.
A noteworthy observation is that considerable variability was noted in the data. For example, some data points appear to be outliers where the EIN increased unexpectedly with luminance intensities (see Figure 5). Furthermore, some participants reported difficulty attending to the peripheral target as they had to suppress the urge to saccade to the target. The effort to suppress saccades over a long period of time could explain the fatigue of some participants and some variability in the results. 
Experiment 2a. Annulus stimulus
The main goal of the current study was to develop a psychophysical paradigm to assess the efficiency of rods. The considerable variability observed in the first experiment for young, healthy adults limits the usefulness of this method, especially for populations for which more variability is expected (e.g., older adults, patients with age-related macular degeneration). Furthermore, some participants reported fatigue and difficulty maintaining fixation while attending to a peripheral stimulus, which could also limit the use of this method for populations having greater difficulty maintaining fixation while attending to a peripheral stimulus to which they must suppress the urge to saccade (Butler, Zacks, & Henderson, 1999). The main aim of the second experiment was to find a paradigm providing more stable results. To facilitate fixation, an annulus centered on fixation was used as a target so that the participants gazed at its center (Figure 7). Furthermore, because the fixation point was difficult to see under scotopic conditions, a black annulus larger than the stimulus was added to indicate to the participant where to gaze when the fixation point was not visible because of the central scotoma under scotopic conditions. Importantly, using an annulus centered on fixation as a target considerably reduced the participants’ urge to saccade to the peripheral stimulus compared to the first experiment. 
Figure 7.
 
Stimuli examples presented on a red display without visual noise (left) and on a blue display with noise (right) during the second experiment. The black dot in the middle of the annulus was the fixation point, and the larger black annulus helped maintain central fixation when the central fixation point was not visible under scotopic conditions.
Figure 7.
 
Stimuli examples presented on a red display without visual noise (left) and on a blue display with noise (right) during the second experiment. The black dot in the middle of the annulus was the fixation point, and the larger black annulus helped maintain central fixation when the central fixation point was not visible under scotopic conditions.
In the first experiment, different temporal frequencies were tested with only one spatial frequency. Because photon noise is expected to be spatially and temporally white, different spatial and temporal frequencies were tested in the second experiment. 
Method
Participants
Four healthy young adults between the ages of 26 and 33 years who did not participate in the first experiment participated in the second experiment. All participants respected the same inclusion/exclusion criteria as in the first experiment. 
Stimuli
The target was a radial sine-wave grating (Figure 7) rotating either clockwise or counterclockwise, and participants were asked to discriminate its rotation direction. The spatial window of the target was an annulus sine-wave grating with a radius from 9° to 11° plus smooth edges following a half-cosine of 1°. The temporal window lasted for 500 ms plus 125-ms half-cosine on and off ramps. The stimulus was presented in the middle of the screen, and the participant was asked to gaze at the central fixation point. A black annulus with a radius of 15° was also presented to help the participant maintain fixation when the central fixation point was not visible under scotopic conditions. For each luminance intensity, the contrast threshold was measured without visual noise added to the display for two spatial frequencies (8 and 16 cycles per circumference [cpc]), two temporal frequencies (1 and 4 Hz), and two display colors (blue and red). Contrast thresholds with visual noise added to the display were measured only at the highest luminance intensity for each of the eight conditions (2 spatial frequencies × 2 temporal frequencies × 2 display colors). The visual noise added to the display was truncated filtered noise (Jules Étienne et al., 2017) with a low-pass filter with a cutoff frequency of 2 cpd, temporally white (refreshed at 60 Hz), and truncated at 1 SD (i.e., applying a maximum and minimum to noise samples after applying the filter), and its contrast was set at 50%, resulting in a noise energy of 161 µs deg2. Note that the signal spatial frequencies were 8 and 16 cpc (i.e., 0.125 and 0.25 cpd, respectively) and that the noise cutoff spatial frequency was 2 cpd, which is more that 1 octave of the signal spatial frequency. Indeed, the noise filter width known to have an impact on performance is ±1 octave of the signal spatial frequency, and the noise masking impact on performance is not known to increase beyond this range (Jules Étienne et al., 2017; Pelli, 1981; Stromeyer & Julesz, 1972). Nonetheless, we did address this assumption in a control experiment where the impact of the noise cutoff spatial frequency on contrast threshold in noise (i.e., calculation efficiency) was evaluated, which validated this assumption (see Supplementary Material). 
Procedure
The same procedure as in the first experiment was used in the second experiment with regard to measuring visual acuity, measuring pupil size, adapting to light, testing order (ascending luminance intensities, separate sessions for red and blue displays), viewing condition (monocular at 50 cm), and staircase procedure. 
Results and discussion
Figure 8 represents the contrast thresholds in the absence of noise as a function of the photopic luminance intensity for different combinations of spatial and temporal frequencies with red and blue displays. As in the first experiment, contrast thresholds were lower with the blue display compared to the red display at low luminance intensities (<∼5 photopic Td), which suggests a considerable contribution of rods at low luminance intensities with a blue display. 
Figure 8.
 
Contrast thresholds in the absence of noise as a function of photopic luminance intensity for each participant (different graphs) for different combinations of spatial and temporal frequencies (different symbols) tested over red and blue displays (red and blue lines, respectively).
Figure 8.
 
Contrast thresholds in the absence of noise as a function of photopic luminance intensity for each participant (different graphs) for different combinations of spatial and temporal frequencies (different symbols) tested over red and blue displays (red and blue lines, respectively).
The contrast thresholds for noise as a function of the temporal and spatial frequencies with red and blue displays are shown in Figure 9. For each spatiotemporal frequency (8 or 16 cpc, and 1 or 4 Hz), similar contrast thresholds were observed with the blue and red displays, suggesting that the calculation efficiency was relatively independent of the display color, as in the first experiment. The calculation efficiency (k) for each spatial and temporal frequency was therefore assumed to be the same for the two colors and was derived from the geometric mean contrast threshold in noise (Equation 2). 
Figure 9.
 
Contrast thresholds in noise measured when the display was red or blue (respective lines) as a function of temporal frequency. The contrast thresholds in noise measured at 8 and 16 cpc are represent by the solid lines and dotted lines, respectively. The error bars represent the standard error of the mean computed in log units.
Figure 9.
 
Contrast thresholds in noise measured when the display was red or blue (respective lines) as a function of temporal frequency. The contrast thresholds in noise measured at 8 and 16 cpc are represent by the solid lines and dotted lines, respectively. The error bars represent the standard error of the mean computed in log units.
For each spatiotemporal frequency and each display color, the EIN as a function of luminance intensity was derived using Equation 3 based on the calculation efficiency (k) and the contrast thresholds in the absence of noise as a function of luminance intensity (Figure 8). The EIN levels of each participant were found to roughly match the de Vries–Rose law (i.e., a log–log slope of –1) over an intermediate range of luminance intensities for each color displays (see Figure 10 in which the fits closely match the data). 
Figure 10.
 
EIN (derived from the data presented in Figures 8 and 9) as a function of photopic luminance intensity for each participant (different graphs). The de Vries–Rose law is represented by a slope of –1 over the red and blue data (red-fit slope and blue-fit slope, respectively) for each participant.
Figure 10.
 
EIN (derived from the data presented in Figures 8 and 9) as a function of photopic luminance intensity for each participant (different graphs). The de Vries–Rose law is represented by a slope of –1 over the red and blue data (red-fit slope and blue-fit slope, respectively) for each participant.
For each testing condition (two colors, two temporal frequencies, and two spatial frequencies), the photon noise was derived from the EIN by fitting a curve with a slope of –1 (i.e., de Vries–Rose law; see light blue and red lines in Figure 10) using the same fitting procedure as in Experiment 1. Figure 11 shows the photon noise (geometric mean across participants) as a function of temporal and spatial frequencies for the red and blue displays. A repeated-measures 2 × 2 × 2 ANOVA on the photon noise in log units showed a significant large effect of color, F(1,3) = 808.514, p < 0.001, ω2 = 0.989, but no significant effect of spatial frequency, F(1,3) = 0.666, p = 0.474, ω2 = 0, or temporal frequency, F(1,3) = 1.099, p = 0.372, ω2 = 0.011, and no significant interaction was observed. Thus, the photon noise was found to be roughly independent of the spatial and temporal frequencies. Furthermore, in Figure 10, note that, when approaching the de Vries–Rose law, the EIN is roughly independent of the spatial and temporal frequencies, as can be seen with the overlapping curves for each display. In contrast, when the EIN deviated from the de Vries–Rose law at higher luminance intensities, variability was observed between the spatial and temporal frequencies, as can be seen by the curves being very distinct from each other. 
Figure 11.
 
Geometric mean of the photon noise evaluated when the display was red or blue for the two spatial frequencies as a function of temporal frequency. The error bars represent the standard error of the mean calculated in log units.
Figure 11.
 
Geometric mean of the photon noise evaluated when the display was red or blue for the two spatial frequencies as a function of temporal frequency. The error bars represent the standard error of the mean calculated in log units.
The results were drastically different at high luminance intensities as a repeated-measures 2 × 2 × 2 ANOVA of the EIN in log units at the highest luminance intensity (without any filter) showed a significant large effect of temporal frequency, F(1,3) = 11.13, p = 0.045, ω2 = 0.185, and spatial frequency, F(1,3) = 70.24, p = 0.004, ω2 = 0.580 (Figure 12). Although no significant effect of color was found, F(1,3) = 2.47, p = 0.214, ω2 = 0.056, a significant interaction between color and temporal frequency was observed, F(1,3) = 10.11, p = 0.05, ω2 = 0.065. These results show that, at the highest luminance intensity, the EIN considerably varied with color, spatial, and temporal frequency. Furthermore, at high luminance intensities, the slope of the EIN was much flatter than the de Vries–Rose law, as can clearly be seen in Figure 10. Conversely, the slope was much steeper at the very low luminance intensities. 
Figure 12.
 
Geometric mean of the EIN evaluated at the highest luminance intensity when the display was red or blue for the two spatial frequencies as a function of temporal frequency. The error bars represent the standard error of the mean calculated in log units.
Figure 12.
 
Geometric mean of the EIN evaluated at the highest luminance intensity when the display was red or blue for the two spatial frequencies as a function of temporal frequency. The error bars represent the standard error of the mean calculated in log units.
The fitting method used was simple and effective, but it implicitly assumed that a portion of the function followed the de Vries–Rose law. To objectively test if this was the case, an objective fitting procedure was conducted in which a wide variety of models were fitted (see Supplementary Material). A model with photon noise was found to provide a significantly better fit than other models without photon noise. This analysis was also consistent with the current findings that photon noise significantly varied with the display color, but not the spatial and temporal frequency, and late noise varied with temporal frequency. 
In the second experiment, less variability was observed within participants, as can be seen in Figure 10, compared to the variability noted in the first experiment (Figure 5). Moreover, participants did not report any difficulty in maintaining fixation at the center of the display and mentioned that the black annulus was helpful to keep gazing at the central fixation when they could no longer perceive the central black dot under low luminance intensities conditions. 
Experiment 2b. Calculation efficiency as a function of luminance intensity
In the previous experiments, calculation efficiency was measured only at a high luminance intensity and was assumed to be independent of luminance intensity, as contrast thresholds in high visual noise are roughly independent of luminance intensity under photopic conditions (Braham Chaouche et al., 2020). Although calculation efficiency is roughly independent of luminance intensity when vision is mediated by cones, it may differ when vision is mediated by rods. The aim of this third experiment was to evaluate calculation efficiency (i.e., contrast thresholds in high visual noise) as a function of luminance intensity. 
Method
Stimuli
The target had the same properties as the one in Experiment 2a: a radial sine-wave grating rotating either clockwise or counterclockwise with the same spatial and temporal window, and participants were asked to discriminate its rotating direction. For each luminance intensity, contrast threshold at a spatial frequency of 8 cpc and a temporal frequency of 1 Hz over two display colors (blue and red) was measured with and without visual noise added to the display. Unfortunately, the noise used in Experiment 2a had no significant impact on contrast thresholds at low luminance intensities, so a stronger visual noise was used: truncated filtered noise (Jules Étienne et al., 2017) with a low-pass filter with a cutoff frequency of 1 cpd that was temporally white (refreshed at 30 Hz) and truncated at 1 SD (i.e., applying a maximum and minimum to noise samples after applying the filter), and its contrast was set at 50%, resulting in a noise energy of 1287 µs deg2
Procedure
The procedure was similar to the one in Experiment 2a. Visual acuity measurement, pupil size measurement at the beginning of each new luminance intensity condition, viewing condition (monocular at 50 cm), and the staircase procedure were the same. Contrast thresholds with and without external noise were measured over three luminance intensities: 0.00158, 0.05, and 5 photopic cd/m2 (corresponding to 0.00322, 0.0322, and 0.322 scotopic cd/m2, respectively, for the red display and 0.217, 2.17, and 21.7 scotopic cd/m2, respectively, for the blue display). Each of these six conditions was tested five times. Each participant underwent two experimental sessions over separate days: Contrast thresholds under the two lower luminance intensities conditions were measured during the first session, and contrast thresholds under the highest luminance intensity were measured in the second session. In the first session, participants adapted for 15 minutes to the intermediate luminance intensity condition before the beginning of the experiment, and pupil size was measured with the same procedure as in the first two experiments. Ten contrast thresholds were measured over the red display (five with visual noise added to the display and five without, presented in a random order). Afterward, 10 contrast thresholds were measured over the blue display following the same procedure. After testing to the intermediate luminance intensity, participants adapted for 15 minutes to the darkest luminance intensity and performed the same procedure. In the second session, 10 contrast thresholds were measured based on the same procedure, but no adaptation was required because contrast thresholds were measured at the highest luminance intensity. 
Results and discussion
Contrast thresholds in the presence and absence of visual noise as a function of the photopic luminance intensity for each participant with red and blue displays are presented in Figure 13. As in Experiments 1 and 2a, contrast thresholds in the absence of noise were lower with the blue display compared to the red display at low luminance intensities, which suggests a considerable contribution of rods at low luminance intensities with a blue display. 
Figure 13.
 
Contrast thresholds (geometric mean over five measurements) in the presence and absence of noise as a function of photopic luminance intensity for each participant (different graphs) tested over red and blue displays (red and blue lines, respectively). Note that participant P4 was not sensitive enough to perceive the stimulus at the maximum contrast at the lowest luminance intensity over the red display (both with and without noise), which is marked as 1.
Figure 13.
 
Contrast thresholds (geometric mean over five measurements) in the presence and absence of noise as a function of photopic luminance intensity for each participant (different graphs) tested over red and blue displays (red and blue lines, respectively). Note that participant P4 was not sensitive enough to perceive the stimulus at the maximum contrast at the lowest luminance intensity over the red display (both with and without noise), which is marked as 1.
As in the previous experiments, contrast thresholds limited by visual noise (and thereby calculation efficiency; see Equation 2) were similar with the red and blue displays. Note that the visual noise did not considerably affect contrast thresholds (i.e., similar threshold with and without noise) for the red display at the lowest luminance intensity, suggesting that the contrast threshold was not limited by the visual noise and calculation efficiency cannot be evaluated under this condition. With the blue display, however, contrast threshold was consistently greater in the presence of visual noise than in absence of noise. Thus, the calculation efficiency could be evaluated from the contrast thresholds in the presence of visual noise (Equation 2) with the blue display at all luminance intensities. 
More importantly, similar contrast thresholds in the presence of visual noise were observed with the blue display over all luminance intensities, suggesting that the calculation efficiency was relatively independent of luminance intensity, as assumed in the previous experiments. 
General discussion
In the first experiment, the EIN was evaluated as a function of temporal frequency, luminance intensity, and display color (blue and red) using a localized peripheral stimulus. Over the range of luminance intensities at which the de Vries–Rose law was observed (i.e., EIN inversely proportional to the luminance intensity), the EIN was roughly independent of temporal frequencies. However, attending to a localized peripheral stimulus for a prolonged duration was found to be attentively demanding, and considerable variability was observed in the data. In the second experiment, the EIN was measured using an annulus centered on fixation to reduce the urge to gaze at the peripheral stimulus. For the conditions at which the de Vries–Rose law was observed, the EIN was roughly independent of spatial and temporal frequencies. Participants reported less urge to gaze at the peripheral stimulus, and the variability in the data was lower. 
In the second experiment, photon noise was undoubtedly not the main noise limiting contrast sensitivity at high luminance intensities, as the properties of the EIN did not match the expected properties of photon noise. First, the EIN was not inversely proportional to luminance intensity (i.e., de Vries–Rose law) (Pelli, 1990; Silvestre et al., 2018), as the log–log slopes flattened with increasing luminance intensity asymptotically approaching Weber's law (slope of 0). In addition, the EIN considerably varied with spatial and temporal frequency, whereas photon noise was expected to be spatially and temporally white (i.e., photon detection spatiotemporally uncorrelated) (Pelli, 1990). Thus, the properties observed at high luminance intensities are not compatible with the expected properties of EIN when contrast sensitivity is limited by photon noise (i.e., following the de Vries–Rose law and being spatiotemporally white). On the other hand, these properties (EIN varying little with luminance intensity and varying considerably with spatial and temporal frequency) are consistent with contrast sensitivity being limited by late noise, which typically limits contrast sensitivity at high luminance intensities (Allard & Arleo, 2017; Silvestre et al., 2018
At low luminance intensities in the second experiment, the main noise limiting contrast sensitivity was also undoubtedly not photon noise, as the EIN was not inversely proportional to luminance intensity (i.e., de Vries–Rose law). The much steeper log–log slopes could be explained by contrast sensitivity being limited by early noise (i.e., linear law) (Silvestre et al., 2018), which is the typical internal noise limiting contrast sensitivity at low luminance intensities. 
On the other hand, the EIN was found to match the properties of photon noise over a wide range of conditions: intermediate luminance intensities in Experiment 2a, all luminance intensities in red in Experiment 1, and low luminance intensities in blue in Experiment 1. Under these conditions, the de Vries–Rose law was observed (i.e., EIN roughly inversely proportional to luminance intensity), and the EIN did not considerably vary with spatial and temporal frequencies, which is expected to be spatially and temporally white (Graham & Hood, 1992; Pelli, 1990). These results are consistent with contrast sensitivity being limited by photon noise. 
Note that, if photon noise plays a considerable role in limiting sensitivity under some conditions, then there must be other sources of noise to explain the linear law (i.e., sensitivity proportional to luminance intensity) and Weber's law (i.e., sensitivity independent of luminance intensity) (Graham & Hood, 1992). Furthermore, if photon noise is the limiting factor under some conditions, there must be another suboptimal factor to explain the non-ideal performance (e.g., suboptimal calculation efficiency) (Geisler, 2003; Geisler, 2011; Graham & Hood, 1992). The model used in the present study (Silvestre et al., 2018) has factors consistent with these constraints (i.e., early noise, late noise, and calculation efficiency), which suggests that photon noise could be a limiting factor when the de Vries–Rose law was observed. 
Nevertheless, the fact that the data roughly followed the de Vries–Rose law does not necessarily indicate that photon noise was a limiting factor; there could be another internal factor explaining the de Vries–Rose law. Furthermore, although contrast sensitivity as a function of luminance intensity is known to follow one of three laws (Linear, de Vries–Rose, and Weber) (Aguilar & Stiles, 1954; Davson, 1990; Díez-Ajenjo & Capilla, 2010; Silvestre et al., 2018), depending on the conditions, this does not necessarily suggest that the de Vries–Rose law was observed when it was the law that best fitted the data. We cannot rule out the possibility that the data observed followed another unknown law that was closer to the de Vries–Rose law than the other two laws. However, the fact that the EIN was independent of the temporal frequency also suggests that photon noise was the limiting noise source. Indeed, the temporal integration of photoreceptors (which behaves as a temporally low-pass filter considerably reducing the contrast of high temporal frequencies) (Allard & Arleo, 2017) should cause the impact of any source of internal noise occurring after the photoreceptors to exponentially increase with temporal frequency. Indeed, based on physiological data, Kelly and Wilson (1978) showed that the temporal response of photoreceptors would reduce contrast sensitivity by a factor of about 3 from 1 to 4 Hz (value derived from equation 1 in Kelly & Wilson, 1978). Thus, the fact that EIN was found to be independent of the temporal frequency must be due to a noise occurring before the temporal integration of the photoreceptors (i.e., photon noise) or to a late noise source occurring after a contrast gain compensating for the effective contrast reduction with high temporal frequencies occurring at the photoreceptor level. These two opposite contrast gains would have to be opposite and of similar magnitude to result in no net gain. Similarly, the fact that the EIN was independent of the spatial frequency suggests that the net contrast gain before the limiting noise source is roughly independent of the spatial frequency, which could be due to noise occurring early (i.e., before any contrast gain, considerably varying with spatial frequency such as at the bipolar cell level) or late with different contrast gains canceling one another. Given the known existence of photon noise (Braham Chaouche et al., 2021; Geisler, 2003; Geisler, 2011; Pelli, 1990; Silvestre et al., 2018) and the fact that the EIN was independent of the spatial and temporal frequencies when it roughly followed the de Vries–Rose law, the simplest interpretation is that photon noise was the limiting source of noise under these conditions. 
In both experiments, under the conditions in which photon noise was observed (i.e., when the EIN roughly followed the de Vries–Rose law and was independent of the spatial and temporal frequencies), contrast thresholds (and thereby the EIN) for the red display were greater (i.e., worse) than for the blue display of the same photopic luminance intensity, which suggests the implication of rods with the blue display. Indeed, by definition, the same number of photons should be detected by cones for two different colors having the same photopic luminance intensity. Because the photon noise is directly related to the number of photons detected (Braham Chaouche et al., 2021), the photon noise is expected to be the same for the two display colors. The lower photon noise (or lower contrast threshold) with the blue display relative to the red display therefore suggests that the blue display was mediated by rods. These results show that the efficiency of rods to detect light can be evaluated using a blue display. 
Conversely, the implications of cones can be assessed by comparing the photon noise at the same scotopic, rather than photopic, luminance intensity. If both display colors were mediated by rods, then the same photon noise would be expected given equal scotopic luminance intensity (i.e., same number of photons expected to be detected by rods). For equal photopic luminance intensity between the red and blue displays, the scotopic luminance intensity was 67 times greater with the blue display relative to the red display because rods are more sensitive to short wavelengths. Given that the photon noise is inversely proportional to the luminance intensity (de Vries–Rose law), the photon noise at the same photopic luminance intensity (Figures 6 and 11) is expected to be about 67 times greater for the red display relative to the blue display if both were mediated by rods, but the blue–red photon noise ratios were much lower (factors of about 26 and 13 in the first and second experiments, respectively; see Figures 6 and 11, respectively). In other words, at equal scotopic luminance intensity, the photon noise evaluated with the red display was lower (i.e., better) than with the blue, which suggests the implication of cones with the red display. These results show that the efficiency of cones to detect light can be evaluated using a red display. 
Maintaining fixation was obviously easier in the second experiment, presumably because participants could look directly at the center of the target (which reduces the urge to saccade to the peripheral target), and the large black annulus helped them to maintain their gaze on the middle of the screen when the luminance intensity was very low. Indeed, with the rods absent within the foveola (Curcio et al., 1990), the fixation point was not visible in scotopic vision, as vision relies on rods, and the black annulus therefore helped maintain central fixation. These differences could explain the lower within-participant variability observed in the results of the second experiment. 
However, an annulus as a target stimulates photoreceptors over a large area of peripheral retina and thereby measures photon detection over a large retinal area, which differs from a localized target. A localized target obviously measures the efficiency of photoreceptors at the corresponding retinal location. On the other hand, using a large target stimulates a wide retinal area, and it is unclear which retinal location the participants is using to perform the task. Given that the participant probably bases his decision on the most sensitive area being stimulated, the efficiency evaluated is likely the retinal area with the best efficiency stimulated by the target. For example, if a participant has a retinal lesion that causes a scotoma within a certain region of his visual field, the participant could still have a high efficiency if he bases his decision on healthy retinal areas also stimulated by the target. Thus, a large target could be used to evaluate a diffuse deficit affecting all of the retinal areas covered by the target, whereas a localized target would be preferred to test the efficiency of photoreceptors in a specific retinal area with the drawback of making fixation more difficult. 
Conclusions
The current study shows that the efficiency of rods to detect photons can be evaluated using a psychophysical paradigm based on the measurement of two contrast thresholds: with and without visual noise. Without noise, the contrast threshold must be measured under conditions in which photon noise is the source of internal noise limiting contrast sensitivity (i.e., in which case the EIN should follow the de Vries–Rose law), and rods are more sensitive than cones (in which case contrast sensitivity should be higher using a blue display than a red display at the same photopic luminance intensity). The current study found that these conditions were met using a blue stimulus for luminance intensities between ∼0.02 and ∼5 photopic Td. This new psychophysical paradigm could be useful for comparing the impact of retinal diseases such as age-related macular degeneration on the efficiency of rods and cones to detect light. 
Acknowledgments
Supported by grants from the Réseau de Recherche en santé de la vision, Fondation Antoine-Turmel, FQRNT, and NSERC to R.A. and grants from École d'optométrie de l'Université de Montréal and Études supérieures et postdoctorales de l'Université de Montréal to G.R. 
Commercial relationships: none. 
Corresponding author: Rémy Allard. 
Address: School of Optometry, Université de Montréal, Montréal, QC, Canada. 
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Figure 1.
 
Stimuli examples presented on the blue display without external noise (top) and the red display with external noise (bottom) during the first experiment. The target was a sine-wave grating, and the participant was asked to gaze at the white fixation point presented at 20° to the left or right of the target, depending on whether the participant viewed the target with his left or right eye, respectively.
Figure 1.
 
Stimuli examples presented on the blue display without external noise (top) and the red display with external noise (bottom) during the first experiment. The target was a sine-wave grating, and the participant was asked to gaze at the white fixation point presented at 20° to the left or right of the target, depending on whether the participant viewed the target with his left or right eye, respectively.
Figure 2.
 
Normalized spectral energy distribution of the blue and red color gun emitted by the display.
Figure 2.
 
Normalized spectral energy distribution of the blue and red color gun emitted by the display.
Figure 3.
 
Contrast thresholds in the absence of noise as a function of photopic luminance intensity for each participant (different graphs) for different temporal frequencies (different symbols) when the display was red or blue (red and blue lines, respectively).
Figure 3.
 
Contrast thresholds in the absence of noise as a function of photopic luminance intensity for each participant (different graphs) for different temporal frequencies (different symbols) when the display was red or blue (red and blue lines, respectively).
Figure 4.
 
Geometric mean contrast thresholds in noise measured when the display was red or blue (respective lines) as a function of temporal frequency. The error bars represent the standard error of the mean computed in log units.
Figure 4.
 
Geometric mean contrast thresholds in noise measured when the display was red or blue (respective lines) as a function of temporal frequency. The error bars represent the standard error of the mean computed in log units.
Figure 5.
 
EIN (derived from the data presented in Figures 3 and 4) as a function of photopic luminance intensity for each participant (different graphs). The thick lines illustrate the de Vries–Rose law (i.e., log–log slopes of –1), representing the impact of photon noise (see Equation 4) when the display was red or blue (respective colors) for each participant.
Figure 5.
 
EIN (derived from the data presented in Figures 3 and 4) as a function of photopic luminance intensity for each participant (different graphs). The thick lines illustrate the de Vries–Rose law (i.e., log–log slopes of –1), representing the impact of photon noise (see Equation 4) when the display was red or blue (respective colors) for each participant.
Figure 6.
 
Geometric mean photon noise measures when the display was red or blue (respective lines) as a function of temporal frequency. The error bars represent the standard error of the mean computed in log units.
Figure 6.
 
Geometric mean photon noise measures when the display was red or blue (respective lines) as a function of temporal frequency. The error bars represent the standard error of the mean computed in log units.
Figure 7.
 
Stimuli examples presented on a red display without visual noise (left) and on a blue display with noise (right) during the second experiment. The black dot in the middle of the annulus was the fixation point, and the larger black annulus helped maintain central fixation when the central fixation point was not visible under scotopic conditions.
Figure 7.
 
Stimuli examples presented on a red display without visual noise (left) and on a blue display with noise (right) during the second experiment. The black dot in the middle of the annulus was the fixation point, and the larger black annulus helped maintain central fixation when the central fixation point was not visible under scotopic conditions.
Figure 8.
 
Contrast thresholds in the absence of noise as a function of photopic luminance intensity for each participant (different graphs) for different combinations of spatial and temporal frequencies (different symbols) tested over red and blue displays (red and blue lines, respectively).
Figure 8.
 
Contrast thresholds in the absence of noise as a function of photopic luminance intensity for each participant (different graphs) for different combinations of spatial and temporal frequencies (different symbols) tested over red and blue displays (red and blue lines, respectively).
Figure 9.
 
Contrast thresholds in noise measured when the display was red or blue (respective lines) as a function of temporal frequency. The contrast thresholds in noise measured at 8 and 16 cpc are represent by the solid lines and dotted lines, respectively. The error bars represent the standard error of the mean computed in log units.
Figure 9.
 
Contrast thresholds in noise measured when the display was red or blue (respective lines) as a function of temporal frequency. The contrast thresholds in noise measured at 8 and 16 cpc are represent by the solid lines and dotted lines, respectively. The error bars represent the standard error of the mean computed in log units.
Figure 10.
 
EIN (derived from the data presented in Figures 8 and 9) as a function of photopic luminance intensity for each participant (different graphs). The de Vries–Rose law is represented by a slope of –1 over the red and blue data (red-fit slope and blue-fit slope, respectively) for each participant.
Figure 10.
 
EIN (derived from the data presented in Figures 8 and 9) as a function of photopic luminance intensity for each participant (different graphs). The de Vries–Rose law is represented by a slope of –1 over the red and blue data (red-fit slope and blue-fit slope, respectively) for each participant.
Figure 11.
 
Geometric mean of the photon noise evaluated when the display was red or blue for the two spatial frequencies as a function of temporal frequency. The error bars represent the standard error of the mean calculated in log units.
Figure 11.
 
Geometric mean of the photon noise evaluated when the display was red or blue for the two spatial frequencies as a function of temporal frequency. The error bars represent the standard error of the mean calculated in log units.
Figure 12.
 
Geometric mean of the EIN evaluated at the highest luminance intensity when the display was red or blue for the two spatial frequencies as a function of temporal frequency. The error bars represent the standard error of the mean calculated in log units.
Figure 12.
 
Geometric mean of the EIN evaluated at the highest luminance intensity when the display was red or blue for the two spatial frequencies as a function of temporal frequency. The error bars represent the standard error of the mean calculated in log units.
Figure 13.
 
Contrast thresholds (geometric mean over five measurements) in the presence and absence of noise as a function of photopic luminance intensity for each participant (different graphs) tested over red and blue displays (red and blue lines, respectively). Note that participant P4 was not sensitive enough to perceive the stimulus at the maximum contrast at the lowest luminance intensity over the red display (both with and without noise), which is marked as 1.
Figure 13.
 
Contrast thresholds (geometric mean over five measurements) in the presence and absence of noise as a function of photopic luminance intensity for each participant (different graphs) tested over red and blue displays (red and blue lines, respectively). Note that participant P4 was not sensitive enough to perceive the stimulus at the maximum contrast at the lowest luminance intensity over the red display (both with and without noise), which is marked as 1.
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