# Peripheral Flicker Fusion at High Luminance: Beyond the Ferry–Porter Law

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## Abstract

**:**

^{4}Trolands, the data conformed to the Ferry–Porter law with a similar slope, as previously established for this eccentricity; however, at higher intensities, the CFF function flattens and saturates at ~90 Hz for a target size of 5.7 degrees, and at ~100 Hz for a target of 10 degrees of angular size. These experimental results could prove valuable for the design of brighter visual displays and illumination sources that are temporally modulated.

## 1. Introduction

_{10}4 Trolands of retinal illuminance. This linear relationship is maintained even when the spectral composition, size and eccentricity of the stimulus change [1,2,3], although the slope and intercept of the function will vary depending on these factors.

_{10}4 Trolands. If, as predicted by these models, the CFF continues to rise linearly with log illuminance, this would mean that brighter displays would need very high refresh rates to avoid the perception of flicker, and even higher to reduce the visibility of other temporal artifacts. If, on the contrary, the response saturates at higher intensities, there would be no need for increasingly higher refresh rates.

^{2}[9]. This would correspond with retinal illuminance values between log

_{10}4.1 and log

_{10}4.5 Trolands—assuming a pupil diameter of 3 mm and after correcting for the Stiles–Crawford effect [8]. However, prototypes already exist of television screens that can reach up to 10,000 cd/m

^{2}[10,11], which would amount to log

_{10}4.8 Trolands for a 3 mm pupil, and even novel micro-displays with intended applications in virtual and augmented reality, which can reach up to 3 million cd/m

^{2}[12] or log

_{10}7.3 Trolands, exist. Although these high-luminance digital displays are not yet widely used or available to the public, mostly due to cost limitations, we can expect that, as these technologies continue to be developed, they will become more commonplace. This raises the need for extending the experimental data available on the human CFF at high-illuminance values. Furthermore, these results would also be relevant for illumination sources that are temporally modulated, such as LEDs, which are frequently controlled through pulse-width modulation.

_{10}2 Trolands, at which point it saturated and started to decrease. The maximum frequency of flicker detected before saturation reached was between 50 to 60 Hz. Their results in other retinal eccentricities tested were similar, albeit paradoxically, they found that the slope was shallower in the periphery, and saturation was reached at lower illuminances. Indeed, most experimental results preceding the study by Tyler [4] had shown a slower response and lower sensitivity to flicker in the periphery [7,13,14,15]. However, Tyler at al. [2] later showed that with more careful control of the experimental setup and adjusting the size of the target to stimulate a similar number of photoreceptors at different retinal locations, the temporal response was indeed faster and sensitivity to flicker was higher in the periphery.

## 2. Materials and Methods

#### 2.1. Participants

#### 2.2. Apparatus

^{2}. This reduction in voltage changes reduced the rise and fall time of the LEDs significantly, allowing the square waveform to be preserved (see Figure 2) and the total luminance to remain constant regardless of the frequency.

^{2}over the first 5 min, beyond which no further consistent changes were found, with only small random fluctuations around the mean of 23,250 cd/m

^{2}with a standard deviation of 204 cd/m

^{2}. To account for the small decrease at the start, we ensured the experimental setup was turned on for at least 15 min before commencing the data collection.

#### 2.3. Task and Design

#### 2.4. Procedure

#### 2.5. Threshold Estimation

_{ma}

_{x}, is calculated as

_{max}as a function of the period of flicker was obtained, it was fitted with a Quick psychometric function [19] through a maximum likelihood procedure, to obtain the estimates of the threshold, slope and lapse rate (which were allowed to vary, but constrained between 0 and 0.03). Standard errors of the threshold estimates were obtained through parametric bootstrap analysis with 600 simulations. Goodness-of-fit measurements based on the likelihood ratio test were then obtained for all psychometric function fits, and those that were deemed to be poor (p < 0.05) were discarded. This resulted in just one threshold estimate among all subjects being excluded.

_{max}and psychometric function fit.

#### 2.6. Data Analysis

^{2}and d is the pupil diameter in mm.

## 3. Results

**Figure 3.**Example of the results obtained for one participant at one retinal illuminance level. The left panel shows the proportion of hits, as a function of the period of flicker, and the proportion of false alarms. The middle panel shows the calculated d′ as a function of the period of flicker presented. Note that when the proportion of hits is lower than the proportion of false alarms, d′ is set to 0. The right panel shows the unbiased proportion correct (PC

_{max}) as a function of the period of flicker, the Quick psychometric function fitted, and the estimated threshold. In all panels the corresponding frequency of flicker is displayed in the top abscissa.

_{10}retinal illuminance (log I) for each participant in each experimental session are presented in Figure 4. For all subjects, the CFFs were measured for a test field size of 5.7 degrees of visual angle, while with two of the subjects we also collected data for a stimulus diameter of 10 degrees of visual angle. In general, we see that the CFF thresholds rise linearly with log I up to approximately log

_{10}4 Trolands, at which point it saturates between 80 and 90 Hz for the 5.7° stimulus. The larger stimulus diameter of 10°, placed at the same central eccentricity, increased the intercept of the function, and accordingly, the value at which it saturates (~100 to 110 Hz), but shows a similar slope. This means that the absolute sensitivity to flicker increases, but the rate at which it changes with increasing intensity is the same. We can also see that the point at which the response starts saturating is similar as for the smaller stimulus size within the same subject. Overall, there is considerable inter-subject variability in both the slope of the linear portion and the value at which the function asymptotes, as well as intra-subject variability between sessions for some participants.

#### 3.1. Segmented Linear Regression

_{10}4 Trolands, and to facilitate the comparison of results, we chose to firstly fit the data with a piece-wise linear function with two segments. For this, we used the Segmented R library, which allows us to estimate the breakpoint and its confidence intervals directly from the data without making any initial assumptions [21,22]. The results for each individual participant and the average of the parameter estimates are shown in Table 1.

_{10}Trolands, an average of 3.82 for the 5.7° target (mean 95% CI from 3.45 to 4.20 log

_{10}Trolands) and 3.80 for the 10° target (mean 95% CI from 3.47 to 4.13 log

_{10}Trolands). This suggests that the CFF to log illuminance function starts to saturate at an approximately similar value regardless of target size. For one subject, however, the estimated breakpoint is much higher at 4.45 log10 Trolands (95% CI from 4.22 to 4.68 log10 Trolands). This can be observed in Figure 4, where the CFF for this participant continues to increase relatively linearly at intensities where the responses of other observers have already saturated.

_{10}Trolands, when compared to the 5.7° target size (with estimates ranging from −0.19 to 0.42 log

_{10}Trolands), reflecting the increased sensitivity to the same amount of light per unit area of the retina, when more photoreceptors are stimulated.

#### 3.2. Linear Mixed Models

## 4. Discussion

_{10}4 Trolands [2]; however, beyond this intensity, it is unknown if the CFF continues to rise linearly or if saturation is reached. In this study, we aimed to extend the experimental data available on the peripheral CFF at higher light intensity levels than previously reported in the literature. For this, we built an experimental setup following the one described by Tyler et al. [2], with careful control of the illumination source, the stimulus size and location, and the psychophysical method used to estimate the CFF [18].

_{10}4 Trolands, we found that the saturation of the response can start at lower illuminances for some subjects. The estimated breakpoint or sudden change in the rate of the response was between log

_{10}3.6 and 4.4 Trolands among our sample. Beyond this, the response saturates and the rate of increase in the CFF decreases dramatically.

_{10}3.6 and log

_{10}4.6 Trolands, with the CFF reaching just below 90 Hz for the target of 5.7 degrees of visual angle, and approximately 100 Hz for the 10 degrees target. As expected, the thresholds were higher for the test field with larger area, but the speed of the response was the same, which was reflected in higher absolute values of CFF but similar slope estimates. This is anticipated to happen for a stimulus of a larger area but equal retinal eccentricity and wavelength of light. In practical terms, as visual displays tend to have larger sizes than the stimuli used, we can expect the maximum frequency at which saturation occurs to be higher.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Luminous output measurements at four different frequencies. The red dots represent the raw individual measurements, while the black markers represent the average of these measurements at each time point.

**Figure 4.**Estimated CFFs as a function of log

_{10}retinal illuminance. Error bars represent the standard error of the CFFs obtained through parametric bootstrap analysis. Different sessions are represented with different colors. The dotted lines represent CFFs obtained for a stimulus of 5.7 degrees of visual angle, and the continuous lines for a stimulus of 10 degrees of visual angle.

**Figure 5.**Quadratic linear mixed model results of CFF as a function of log

_{10}retinal illuminance, for stimulus sizes of 5.7 (

**left**) and 10 (

**right**) degrees of visual angle. The continuous colored lines represent the model fit and the shaded regions represent the 95% confidence intervals. The individual markers show the estimated thresholds for each subject over several experimental sessions, and the color of the markers represents the participant as indicated by the legend.

**Figure 6.**Quadratic linear mixed model results of the 90% “no flicker” threshold as a function of log

_{10}retinal illuminance, for stimulus sizes of 5.7 (

**left**) and 10 (

**right**) degrees of visual angle. The continuous colored lines represent the model fit and the shaded regions represent the 95% confidence intervals. The individual markers show the estimated thresholds for each subject over several experimental sessions, and the color of the markers represents the participant as indicated by the legend.

**Table 1.**Segmented linear regression results of CFF as a function of log10 retinal illuminance for each size of stimuli and subject. The breakpoint is the value of log I at which the piece-wise linear function separates. The estimated values (Est.), their standard errors (SE), and 95% confidence intervals (CI 95%) are shown. The slopes are in units of Hz/decade, and the x-intercept and breakpoint in units of log10 Trolands.

Target Size/ Subject | Breakpoint [log _{10} Td] | Segment 1 | Segment 2 | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Slope [Hz/decade] | x-Intercept [log _{10} Td] | Slope [Hz/dec.] | |||||||||||

Est. | 95% CI | SE | Est. | 95% CI | SE | Est. | SE | Est. | SE | ||||

5.7° | 1 | 3.61 | 2.85 | 4.37 | 0.4 | 21.4 | 12.9 | 29.8 | 4.2 | 0.36 | 0.47 | 6.3 | 1.1 |

2 | 3.71 | 3.55 | 3.86 | 0.1 | 25.5 | 23.5 | 27.5 | 1.0 | 0.42 | 0.11 | 3.3 | 0.8 | |

3 | 3.73 | 3.51 | 3.96 | 0.1 | 20.7 | 18.6 | 22.7 | 1.0 | −0.01 | 0.14 | 4.9 | 0.9 | |

4 | 4.45 | 4.22 | 4.68 | 0.1 | 18.2 | 16.6 | 19.8 | 0.8 | −0.19 | 0.13 | 1.1 | 1.6 | |

5 | 3.63 | 3.15 | 4.11 | 0.2 | 19.2 | 15.3 | 23.2 | 1.9 | −0.13 | 0.28 | 5.1 | 1.6 | |

Mean | 3.82 | 3.45 | 4.20 | 0.2 | 21.0 | 17.4 | 24.6 | 1.8 | 0.09 | 0.23 | 4.2 | 1.2 | |

10° | 2 | 3.87 | 3.67 | 4.06 | 0.1 | 24.0 | 19.5 | 28.4 | 2.1 | −0.54 | 0.26 | −9.2 | 4.5 |

3 | 3.73 | 3.27 | 4.20 | 0.1 | 19.5 | 15.1 | 23.9 | 1.0 | −0.82 | 0.15 | 6.2 | 1.8 | |

Mean | 3.80 | 3.47 | 4.13 | 0.1 | 21.7 | 17.3 | 26.1 | 1.6 | −0.68 | 0.21 | −1.5 | 3.2 |

**Table 2.**Linear mixed model results of CFF as a function of log10 retinal illuminance and squared log10 retinal illuminance. The parameters estimate (Est.), their standard errors (SE) and 95% confidence intervals (95% CI) are shown, as well as the t-test results and the standard deviation of the different random effects included (RE SD).

5.7° Stimulus—CFF [Hz] | |||||||||
---|---|---|---|---|---|---|---|---|---|

Parameter | Est. | SE | 95% CI | t-Ratio | df | p-Value | RE SD | ||

Subject | Session | ||||||||

Intercept | −26.05 | 3.42 | −32.80 | −19.30 | −7.63 | 157 | <0.001 | 3.03 | 0.86 |

log_{10} I | 39.86 | 1.71 | 36.49 | 43.24 | 23.34 | 157 | <0.001 | - | - |

(log_{10} I)^{2} | −3.57 | 0.22 | −3.99 | −3.14 | −16.58 | 157 | <0.001 | - | - |

10° Stimulus—CFF [Hz] | |||||||||

Parameter | Est. | SE | 95% CI | t-Ratio | df | p-Value | RE SD | ||

Session | |||||||||

Intercept | −70.52 | 14.03 | −99.26 | −41.77 | −5.03 | 28 | <0.001 | 4.05 | |

log_{10} I | 80.80 | 8.38 | 63.64 | 97.97 | 9.64 | 28 | <0.001 | - | |

(log_{10} I)^{2} | −9.59 | 1.21 | −12.07 | −7.11 | −7.92 | 28 | <0.001 | - |

**Table 3.**Linear mixed model results of the 90% “no flicker” threshold as a function of log10 retinal illuminance and squared log10 retinal illuminance. The parameters estimate (Est.), their standard errors (SE) and 95% confidence intervals (95% CI) are shown, as well as the t-test results and the standard deviation of the different random effects included (RE SD).

5.7° Stimulus—90% “No Flicker” Threshold [Hz] | |||||||||
---|---|---|---|---|---|---|---|---|---|

Parameter | Est. | SE | 95% CI | t-Ratio | df | p-Value | RE SD | ||

Subject | Session | ||||||||

Intercept | −25.64 | 5.19 | −35.89 | −15.39 | −4.94 | 153 | <0.001 | 4.28 | 1.25 |

log_{10} I | 43.34 | 2.64 | 38.13 | 48.55 | 16.44 | 153 | <0.001 | - | - |

(log_{10} I)^{2} | −3.81 | 0.33 | −4.47 | −3.15 | −11.46 | 153 | <0.001 | - | - |

10° stimulus—90% “no flicker” threshold [Hz] | |||||||||

Parameter | Est. | SE | 95% CI | t-Ratio | df | p-Value | RE SD | ||

Session | |||||||||

Intercept | −68.72 | 28.14 | −126.67 | −10.77 | −2.44 | 25 | 0.022 | 5.37 | |

log_{10} I | 84.82 | 17.07 | 49.66 | 119.98 | 4.97 | 25 | <0.001 | - | |

(log_{10} I)^{2} | −10.03 | 2.49 | −15.16 | −4.90 | −4.03 | 25 | <0.001 | - |

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**MDPI and ACS Style**

Fernandez-Alonso, M.; Innes, W.; Read, J.C.A.
Peripheral Flicker Fusion at High Luminance: Beyond the Ferry–Porter Law. *Vision* **2023**, *7*, 26.
https://doi.org/10.3390/vision7010026

**AMA Style**

Fernandez-Alonso M, Innes W, Read JCA.
Peripheral Flicker Fusion at High Luminance: Beyond the Ferry–Porter Law. *Vision*. 2023; 7(1):26.
https://doi.org/10.3390/vision7010026

**Chicago/Turabian Style**

Fernandez-Alonso, Maydel, Will Innes, and Jenny C. A. Read.
2023. "Peripheral Flicker Fusion at High Luminance: Beyond the Ferry–Porter Law" *Vision* 7, no. 1: 26.
https://doi.org/10.3390/vision7010026