On Modelling the Human Pupil Light Response and Developing a Deep Learning-Driven Spectral Optimisation Framework to Simulate the Metameric Limits of the Melanopic Stimulus Space
The human pupil diameter is of interest in lighting-related research due to its role in regulating the retinal irradiance and its impact on the visual acuity. In the field of cognitive psychology, transient pupil diameter changes could be used to assess physiological body states, such as the cognitive workload, sleepiness or arousal. Technical concepts are also being explored in medical diagnostics, aiming to leverage temporal pupil size changes as a non-invasive biomarker, e.g., in pain monitoring to regulate the administration of anaesthetics or the early detection of neurodegenerative diseases such as Alzheimer’s. Until the turn of the millennium, it was widely assumed that the pupillary light reflex is controlled by the same type of photoreceptors responsible for visual processing. However, recent studies revealed that the afferent pupil control pathway is affected by an interaction of intrinsically photosensitive retinal ganglion cells (ipRGCs) and photoreceptors in the outer retina (cones and rods). Thus, the spectral dependence of the pupil cannot be described by the luminous efficiency function V(λ) alone. Although the last two decades of neurophysiological research revealed new insights into how the pupil control works, empirical pupil models remained largely unchanged. Hence, they are time-invariant and can only predict the pupil diameter as a function of a V(λ)-weighted quantity. In fact, after more than 100 years of pupil light response research, still no unified model exists that could predict the temporal pupil diameter as a function of distinct light spectra. Therefore, one main objective of this work is to develop an approach to model the time- and spectral-dependent behaviour of the afferent pupil control pathway, allowing temporal pupil diameter predictions as a function of photometric quantities in the future.
In this work, an end-to-end measurement setup consisting of a temperature-controlled 15-channel LED luminaire for generating light spectra and a pupillometry system for empirically collecting human pupil size data was developed for investigating the pupil’s light response. In addition, a novel spectral optimisation method is presented, useful to control multi-channel LED luminaires and to engineer arbitrary polychromatic spectra from photometric quantities whose optimised spectra could be used in pupil examinations. In terms of computation time, the presented metameric optimisation method is by a factor of ∼32 (113.8 ± SD 74 optimised spectra per second) faster than the genetic algorithm (3.6 ± SD 0.8 optimised spectra per second), a method that is recommended in the literature.
A total of 490 000 metameric spectra were optimised for 561 chromaticity coordinates in the CIEu’v’- 1976 colour space along the Planckian locus (2700 K to 7443 K, Duv 0 to ±0.048 in Duv steps of ±0. 003) with the developed spectral optimisation method to determine the extent to which the melanopic illuminance of a light spectrum could be varied while leaving the photopic illuminance (Ev = 250 lx) and chromaticity (∆u′, ∆v′ ≤ 0.001) unchanged. Metameric spectra could be applied, for example, to affect the pupil size or the human’s circadian system in interior lighting systems without altering the visual appearance of the illuminated environment concerning chromaticity and (il)luminance. The larger the melanopic contrast between two metameric spectra, the more a non-visual responses could be varied. Previous works in the literature tended to leverage a lower number of metameric spectra for a limited chromaticity range to analyse the capabilities of metamerism, e.g., a recent study analysed six metameric spectra for three target chromaticity coordinates. Therefore, the built database of optimised metameric spectra and the scale of analysis conducted in this work can be considered the most comprehensive in the science of spectral optimisation. Based on the optimised spectra, it was found that the maximum reachable melanopic Michelson contrast ranges between 0.16 and 0.18 if metameric spectra are considered that feature a colour fidelity index of Rf ≥ 85 and Rf,h1 ≥ 85. For example, with a melanopic Michelson contrast of 0.16, the melanopic illuminances could be varied from 135 lx to 185 lx without altering the photopic illuminance (Ev = 250 lx) or chromaticity coordinate (∆u′, ∆v′ ≤ 0.001) of the metameric light spectra. For each used chromaticity target, the upper and lower limits of the melanopic illuminance were identified while keeping the photopic illuminance steady (Ev = 250 lx). Then, the metameric limits of the melanopic stimulus space were mapped into a colour space, capable of indicating the maximum achievable melanopic contrast at steady photopic illuminance (Ev = 250 lx) for each chromaticity coordinate (∆u′, ∆v′ ≤ 0.001). Such a map might be useful for upcoming spectral optimisation tasks to specifically identify chromaticity locations where the highest melanopic contrasts via metameric spectra could be reached without affecting the visual appearance concerning chromaticity and (il)luminance.
In terms of the pupil modelling topic, a literature review was conducted, revealing that eight relevant time-invariant pupil formulas have been proposed from 1926 to 2012, which can predict the equilibrium- state pupil size using a V(λ)-based photometric quantity. Therefore, a benchmarking was performed as a first step to determine the prediction accuracy of three selected luminance-based pupil models (Crawford model, De Groot & Gebhard model, Watson & Yellott model). It was found that for white light spectra with correlated colour temperatures (CCTs) between 2000 K and 10 000 K (L ≈ 100 cd/m2), the pupil models’ prediction errors are within a pre-defined tolerance range of ±0.5 mm when considering the equilibrium-state pupil size. Therefore, with longer light exposures (60 to 300 seconds), it could be possible to empirically describe (approximation) the spectral-dependent sustained pupil size using the luminance. The results indicate that when using the tested white light stimuli with a steady luminance of ∼100 cd/m2, the pupil models’ lack of time dependence might be a more significant source of error than the missing consideration of ipRGCs since the prediction of the short-term pupil light response (one second after light exposure) yields a deviation of 0.71 mm ± SD 0.15 mm (Watson & Yellott model). If chromatic spectra (peak wavelengths: 450nm, 530nm, 610nm, 660nm) at a steady luminance of ∼100 cd/m2 are used to trigger the pupil light response, however, the prediction error of the tested V(λ)-based pupil models could reach about 1.21 mm for the equilibrium-state pupil size.
As an alternative to the existing empirical V(λ)-based pupil models, a novel modelling approach using feed-forward neural networks was developed, allowing to predict the temporal pupil size in response to chromatic (L ≈ 100 cd/m2, peak wavelengths: 450 nm, 530 nm, 610 nm, 660 nm) and polychromatic spectra (L ≈ 100 cd/m2, CCTs: 2007 K, 4983 K, 10 138 K) with a mean absolute error below 0.1 mm. The method allows the reconstruction of the pupil’s temporal behaviour as a function of distinct lighting metrics for the first time. However, the prediction space of the introduced modelling approach is currently limited to the measured data, which were collected using a steady luminance of ∼100 cd/m.
Further, for validating the modelling approach, the training dataset was used, as the methodological development of a pupil modelling approach was the focus of this work. As a next crucial step, the model’s prediction accuracy needs to be validated more extensively using pupil size data that are not applied during the training of the neural networks. Thus, the proposed deep learning-based modelling approach is, in its current state, not capable nor intended to replace existing V(λ)-based pupil models due to the missing validation and limited prediction space. However, due to the integrated neural networks, it is hypothesised that the prediction space could be further generalised as the amount of pupil size data increases in the future. More data must be obtained empirically to face this topic. Compared to the methodological approach of previously published luminance-based pupil formulas, the proposed deep learning-based pupil modelling method could account for adaptive receptor weighting, reconstruct the entire temporal pupil light response, and perhaps pave the way for a unified model of the afferent pupil control pathway in the future
