Abstract
Pupillometry  the study of temporal changes in pupil diameter as a function of external light stimuli or cognitive processing  requires the accurate and gazeangle independent measurement of pupil dilation. Expected response amplitudes often are only a few percent relative to a prestimulus baseline, thus demanding for submillimeter accuracy. Videobased approaches to pupilsize measurement aim at inferring pupil dilation from eye images alone. Eyeball rotation in relation to the recording camera as well as optical effects due to refraction at corneal interfaces can, however, induce socalled pupil foreshortening errors (PFE), i.e. systematic gazeangle dependent changes of apparent pupil size that are on a par with typical response amplitudes. While PFE and options for its correction have been discussed for remote eye trackers, for headmounted eye trackers such an assessment is still lacking. In this work, we therefore gauge the extent of PFE in three measurement techniques, all based on eye images recorded with a single neareye camera. We present both real world experimental data as well as results obtained on synthetically generated eye images. We discuss PFE effects at three different levels of data aggregation: the sample, subject, and population level. In particular, we show that a recently proposed refractionaware approach employing a mathematical 3D eye model is successful in providing pupilsize measurements which are gazeangle independent at the population level.
Introduction
Pupil size and its variation over time have long been recognized as powerful noninvasive metrics correlating with human cognitive processing (Mathôt, 2018). Today, pupillometry is an established research and diagnostic tool with applications in psychology (Laeng et al., 2012), neurology (Laeng & Alnaes, 2019), and medicine (Phillips et al., 2019).
Handheld pupillometers are readily available and routinely used in a clinical setting, e.g. for monitoring the pupillary lightreflex of patients. Other use cases, in particular in psychology and the behavioral sciences, are often better served by employing videobased eyetracking systems, which next to gaze direction, also provide estimates of pupil size (Hutton, 2019). While remote eye trackers record the eyes of the subject using a stationary camera at a distance (e.g. mounted to a computer screen), headmounted eye trackers feature neareye cameras recording the eyes from closeup. Headmounted eye trackers, in particular, hold the promise of giving access to pupilsize signals for subjects which are moving freely in realworld environments.
Videobased approaches to pupillometry estimate pupil size based on a single or a series of eye images. The 3D pupil, i.e. the ocular opening in the center of the iris, in humans is approximately circular (deviations from circularity are discussed e.g. in Wyatt 1995). When a subject is recorded by a camera from an oblique angle, the resulting 2D pupil image, however, is close in shape to an ellipse^{Footnote 1}. At least three optical effects influence apparent pupil shape in a camera image:

(i) Perspective foreshortening  Moving the 3D pupil away from the camera makes it appear smaller.

(ii) Foreshortening with gaze angle  Tilting the 3D pupil relative to the camera makes it appear more elliptic.

(iii) Corneal refraction  Bending of light rays, occurring naturally at corneal interfaces, magnifies and distorts the image of the 3D pupil in a nonlinear fashion.
In the context of pupillometry, the combined effect of (i)(iii) is often referred to as pupil foreshortening error (PFE), where “error” refers to the fact that any of these three factors, when not appropriately accounted for, can lead to incorrect pupilsize estimates.
To illustrate the detrimental effect of PFE on pupillometry results, consider pupil size is estimated by measuring the area of the apparent 2D pupil in image space. Furthermore, assume an eye with constant pupil size, which is recorded by a stationary camera in close proximity to the eye (see Fig. 1A). Due to foreshortening with gaze angle, a rotation of the eye away from the camera results in a decrease of apparent pupil area and thus of estimated pupil size (see Fig. 1B). In the extreme case of recording the eye from a sufficiently oblique view, apparent pupil area altogether reduces to zero. Thus, due to PFE, pupilsize estimates based on apparent pupil area exhibit a pronounced dependency on gaze angle. Note, the exact dependency is also shaped by (i) and (iii).
One option to minimize the impact of PFE is to design experiments which only employ relatively small static gaze targets appearing always at the same position within the test subject’s fieldofview. In such a scenario, relative pupilsize measurements can still be used to assess pupillary dynamics. Many pupillometry experiments, however, do necessitate dynamically changing gaze directions, as e.g. studies of visual search and reading tasks. Indeed, in these cases pupilsize estimates based on apparent pupil area can introduce errors comparable in magnitude to the cognitively induced pupilsize changes to be measured (Gagl et al., 2011). It is thus desirable to develop approaches to the measurement of pupil size, which do not exhibit any systematic variation with gaze angle.
A perfect measurement method would generate pupilsize measurements bearing no correlation with gaze angle for any given eye image, i.e. on the sample level. Given the variability in eye physiology and the number of simplifying assumptions necessary for deriving pupil size from camera images, at present such a tool appears out of reach. Pupillometry experiments, however, often involve extensive averaging over repeated trials of the same experiment (Sirois and Brisson, 2014; Laeng & Alnaes, 2019), either with the same subject or a population of subjects. Devising measurement methods which are independent of gaze angle on average, either on the subject level and/or the population level, thus is a relevant and pressing pursuit.
Several pioneering works have raised awareness for PFEbased confounds in pupillometry research using remote eye trackers and have proposed strategies for correcting for gazeangle induced artifacts (Gagl et al., 2011; Brisson et al., 2013; Hayes & Petrov, 2016). Even though Pomplun and Sunkara (2003) have shared first insights on PFE in the headmounted case, an indepth analysis for this scenario is still lacking. We close this gap and shed light on the effect of PFE on pupilsize measurements by means of headmounted eye trackers, more specifically, of the widely used Pupil Core headset sold by Pupil Labs (Kassner et al., 2014). Our contributions include the following:

We present results of an experimental study, designed to gauge the extent of gazeangle dependency in three approaches to pupilsize measurement.

We provide experimental data, indicating gazeangle dependency is critically shaped by subjectspecific factors.

In particular, we show that the refractionaware approach proposed by Dierkes et al., (2019) successfully eliminates gazeangle dependencies at the population level.

In the framework of a simulation study, we show that the results of our experimental study are in quantitative agreement with theoretical predictions.
Related work
The present study is related to (i) experimental and theoretical work with regard to the effects of ocular optics on apparent pupil shape as recorded by neareye cameras and (ii) techniques proposed for assessing and correcting gazeangle dependent errors of pupilsize measurements in remote and headmounted eye trackers.
Apparent pupil shape
Early experimental quantification of apparent pupil shape as a function of viewing angle has been contributed by Spring and Stiles (1948) and Jay (1962). For a more recent example, see the work by Mathur et al., (2013). Making use of photographic techniques, these studies map out the systematic changes in geometric properties of the apparent pupil when viewing a 3D pupil of constant radius at varying angles. Their data shows that apparent pupil shape can not be accounted for by assuming a mere perspective projection of the 3D pupil circle into the image plane. In particular, the rate of pupil foreshortening was consistently found to be lower than implied by this simplifying assumption. Instead, apparent pupil shape critically depends on effects introduced by refraction of light rays at corneal interfaces. From a theoretical perspective, the repercussions of corneal refraction on apparent pupil shape have been investigated by Fedtke et al., (2010) and Aguirre (2019). Employing a raytracing approach in the framework of the Navarro eye model (Navarro et al., 1985), these studies demonstrate that the experimental data on apparent pupil shape can be largely accounted for when refraction at corneal interfaces is considered. While Fedtke et al. furthermore provide an indepth analysis of the 3D shape of the socalled entrance pupil, Aguirre presents arguments for resolving remaining discrepancies between theory and experimental observation by means of introducing noncircular 3D pupils into his theory. For an experimental study analyzing the noncircularity of the 3D pupil see the work by Wyatt (1995).
All works mentioned provide fundamental insights as to how for a given 3D pupil, ocular optics determines apparent pupil shape as recorded by a neareye camera. They do not, however, furnish any means for solving the inverse problem, i.e. for inferring pupil size from given eye images. It is the latter question that we are addressing in the present study.
Gazeangle dependency of pupilsize measurements
In a review, Sirois and Brisson (2014) claim that gazeangle induced pupilsize “errors are systematic, and relatively easy to assess and correct”. While errors are certainly systematic, they depend in a complex way on the pupil radius itself, the particular imaging geometry, and individual physiological parameters of the test subject’s eyes. This makes their assessment in a realworld setting a challenging and laborious task. This becomes apparent from the work of several groups (Gagl et al., 2011; Brisson et al., 2013; Hayes & Petrov, 2016), which all have proposed ways for mitigating gazeangle induced pupilsize measurement errors in commercially available remote eyetracking systems.
Gagl et al., (2011) studied the dependence of the pupilsize output of the SRResearch Eyelink 1000 remote eye tracker on horizontal gaze angle. To this end, the authors recorded pupilsize data from (i) test persons performing an effortless zstring reading task and (ii) an artificial eye with constant pupil radius that was rotated horizontally to mimic the horizontal eye movements occurring in (i). In both sets of experiments, pupil size was assumed constant over time. By measuring consistent systematic deviations in both cases, the authors showed that pupilsize estimates obtained with the Eyelink 1000 are indeed prone to gazeangle induced artifacts. As a remedy, Gagl et al. proposed correction functions based on polynomial leastsquare fits to the data obtained in (i) and (ii). They further argued for the efficacy of their approach by successfully correcting a third set of pupilsize data, this time recorded during an effortful sentence reading task. More specifically, they could show that only after applying their correction functions, their data was consistent with findings reported in the literature for similar cognitive tasks.
Brisson et al., (2013) investigated the effect of both horizontal and vertical gaze direction on pupilsize estimation using three remote eyetracking systems (Tobii T120/X120, Eyelink 1000). They recorded data from test subjects performing an effortless task, consisting of the pursuit of a dot describing elliptical movement on a display screen. Given the even illumination and the negligible cognitive load induced, they assumed an approximately constant pupil size. Systematic deviations correlating with screen position were found in the measurements by all three systems. Brisson et al. further showed that 1020 % of the observed pupilsize variation could be explained by a linear regression of pupil size from horizontal and vertical screen coordinates, with “the remaining variation [being] intra and interindividual variation”.
Hayes and Petrov (2016) took a similar route as Gagl et al., in that they mapped gazeangle dependency of pupilradius estimates utilizing artificial eyes without cornea. Working with the EyeLink 1000, they extended earlier results by employing three artificial eyes, each with a different fixed pupil size. In addition, they systematically measured deviations in estimated pupil size as a function of gaze position across three experimental layouts, varying the relative distances between the eyetracking camera, eye, and display. Making simplifying assumptions specific to the remote case, e.g. that eyeball diameter is negligible relative to the distance between eye and recording camera, they derived a geometric model of PFE for eyes without cornea. They showed that their model is able to reduce the root mean squared error of pupil size measurements of the artificial eyes by 82.5 % when the model parameters were preset to the physical layout dimensions, and by 97.5 % when numerically optimizing the model parameters to fit the measured pupil size errors. Finally, they suggested to incorporate empirical foreshortening functions as measured on dilated human eyes by Mathur et al., (2013) into their model to account for refraction effects in real eyes.
Each of the seminal works outlined above  which have also been discussed by Mathôt et al., (2018)  has devised ways for generating phenomenological correction functions to at least partially eliminate gazeangle induced errors in pupilsize measurements. Each has certain merits and detriments. While the use of artificial eyes offers the possibility of generating measurements for known groundtruth pupil sizes, artificial eyes also tend to be anatomically crude. In particular, in case they lack an optically realistic cornea they will not reflect contributions of corneal refraction to PFE. Artificial eyes comprising elements mimicking the human cornea to some extent are commercially available, but usually designed for gaze estimation instead of pupillometry quality assurance. Even the more complex ones lack adjustable pupil size and refractive properties of the materials used for construction do not necessarily closely match physiological values (Wang et al., 2017). Mapping gazeangle dependencies with human subjects in scenarios that approximate constant pupil size over time, potentially circumvents this problem. However, at the cost of an unknown groundtruth pupil size, limiting potential correction schemes to being relative multipliers only. In both cases, correction functions are specific to a certain experimental setup, necessitating new measurements whenever the setup is changed. Most relevant for the current study, it is questionable whether the assumptions and approximations made for the remote case port to the headmounted scenario.
The headmounted case was first studied by Pomplun and Sunkara (2003). Employing an EyeLinkII system, they were able to show that the areabased pupilsize estimate provided by the eye tracker was indeed subject to a systematic gazedependent variation (cf. our simulation data shown in Fig. 1B). The authors proposed a calibration routine for correcting for the observed dependency in a personspecific manner. They argued for the efficacy of their approach in a second set of experiments, in which an increase in cognitive load during a series of screenbased tasks could be correctly detected on the basis of gazeangle adjusted pupil dilations.
In more recent work, Dierkes et al. (2018, 2019) study the effect of gaze angle on pupilsize estimates employing a Pupil Core eye tracker as available from Pupil Labs. Taking a different approach, the authors made use of synthetic eye images in order to map the gazeangle dependency of pupilradius estimates as provided by Pupil Capture, the opensource software used to operate the headset. Pupil Capture employs a modelbased approach to gaze and pupilsize estimation which is closely based on the work by Świrski and Dodgson (2013). By means of a raytracing pipeline, Dierkes et al. generated eye images within the framework of the LeGrand model (Le Grand, 1957), a widely used description of the average human eye. In particular, this allowed for accounting for corneal refraction in a realistic way. Providing such a framework replaces the burden of manual measurements of human and/or artificial eyes with comparably cheap and fast computer simulations, which can be easily rerun for different hardware and imaging/camera setups as well as eye physiologies. Subjecting the simulated images to the pupilsize estimation algorithm, the authors established a mapping between groundtruth pupil size (which is always known in simulations) and measured values. In particular, they found that pupilsize estimates deviated by almost 10% when the eye was rotated away from the camera by about 60^{∘}. Based on the data generated, polynomial correction functions were fit, which subsequently were used to successfully correct for gazeangle dependencies in the synthetic pupilsize data. Preliminary realworld data was presented in Dierkes et al., (2018), qualitatively confirming that the ratio of uncorrected to corrected pupil size followed the trend expected from the synthetic data.
Here, we expand on the work by Dierkes et al. in several ways. By means of controlled experiments, which were designed to realize an approximately constant pupil size over time, we provide a direct quantitative verification of their approach in correcting for gazeangle dependencies at the population level. Unlike the method described by Pomplum and Sunkara, no calibration is needed to achieve this result. We further gauge the method proposed by Dierkes et al. against two other techniques, consisting of (i) estimating pupil size by the major axis of the apparent pupil ellipse and (ii) the pupilsize estimate as provided by the uncorrected Świrski model. In particular, we shed light on the ability of all three methods to provide pupilsize estimates which are independent of gaze angle on the sample, subject, and population level. By means of synthetically generated eye images, we furthermore show that our experimental findings are in quantitative agreement with predictions based on a widely used model of ocular optics.
Methods
In this section, we describe the methods, experimental protocols, and data analysis techniques used in this study. We start by recapitulating the three evaluated methods for measuring pupil size. After providing details with regard to the protocol for recording realworld data in two different constant pupilsize scenarios, we briefly introduce the raytracing pipeline employed for generating synthetic eye images. In particular, we outline the analysis and data aggregation steps performed for extracting pupilsize data both from realworld recordings and synthetic eye images on the sample, subject, and population level.
Pupilsize measurement methods
Videobased pupillometry methods can be classified (i) as to whether they explicitly account for gaze angle and/or corneal refraction, (ii) as to whether they output a physical pupil aperture stop size in [mm] or merely provide a value in arbitrary units, and (iii) as to how many physiological parameters are necessary for their full specification.
Major axis: 2D0p
The first method we consider derives a pupilsize measure directly from the shape of the 2D pupil in a given eye image (see Fig. 2A). The shape of the 2D pupil is commonly approximated by an ellipse. As shown in the Introduction, the area of the pupil ellipse is strongly affected by PFE. A geometric measure expected to depend less strongly on gaze angle is the length of its major axis. The term major axis shall therefore in the following signify the length of the major axis of the pupil ellipse in [pixels], suitably fitted to the region corresponding to the 2D pupil within a given eye image (see examples in Fig. 4).
This imageimmanent 2D method for pupilsize measurement requires no physiological input parameters. We therefore also refer to it as 2D0p. Note, no explicit strategies for accounting for corneal refraction nor for changes in gaze angle are employed by this method.
3D Eye model without cornea: 3D1p
Reporting the length of the major axis of the pupil ellipse results in a pupilsize output in units of [pixels]. As pointed out by several authors, pupillometry experiments should preferably report pupil size in [mm] (see e.g. Beatty and LuceroWagoner (2000), Kelbsch et al., (2019), and Köles (2017)). By fitting a mathematical 3D eye model to video observations of the eye, socalled modelbased approaches allow for deriving pupilsize measures in actual physical units. As a second method for measuring pupil size, we will therefore consider the modelbased approach proposed by Świrski and Dodgson (2013).
Świrski et al. model the eye as comprising an eye sphere of fixed radius, with the 3D pupil being a circle of variable size tangent to it (see Fig. 2B). While changes in gaze angle correspond to rotations of the eye sphere around its center, changes in pupil dilation correspond to changes in the radius of the tangent pupil circle. Given the state of the eye model as well as the pose and intrinsics of a camera, the Świrski model predicts the shape of the pupil ellipse as appearing in an image taken by the camera. To this end, it assumes the 3D pupil circle is mapped to the image via a perspective projection with a pinhole camera. In particular, lacking a cornea, the Świrski model does not account for corneal refraction.
Given a series of pupil ellipse observations under varying gaze angles, the corresponding 3D location of the eye sphere in the coordinate system defined by the recording camera is estimated by solving a nonlinear optimization problem. After the eye sphere is located, the current pupil radius is estimated based on the pupil ellipse in a given eye image as follows. Essentially reversing the imaging operation, in a first step, the pupil ellipse is “unprojected” to find the 3D circle of radius r = 1 mm which under projection to the camera would yield the observed pupil ellipse. In this step a prior measurement of the intrinsics matrix and distortion coefficients of the camera can be employed to correct for center shift (principal point offset) and image distortion, if any. In a second step, the resulting 3D circle is then scaled along the unprojection cone until it lies tangent to the eye sphere estimated before. The scaling factor is the output pupil radius in units of [mm]. Note, the optical axis of the eye is also determined, since it corresponds to the line connecting the center of the scaled 3D circle and the center of the eye sphere.
The size of the eye sphere is the sole physiological parameter which is used in the Świrski model. We therefore also refer to it as 3D1p.
3D Eye model with cornea: 3D4p
While the method by Świrski et al. inherently takes varying gaze angles into account, it does not actively model corneal refraction. Since the image of the pupil is distorted by the refraction effect of the cornea, unprojection of the resulting pupil ellipse does not yield the correct 3D pupil circle. As already indicated in “Gazeangle dependency of pupilsize measurements”, Dierkes et al., (2019) have presented a modelbased technique for determining 3D eyeball position, optical axis, and pupil size from video images, which explicitly corrects for corneal refraction. Their approach constitutes the third method considered in this study.
The main idea underlying the method by Dierkes at al. is to derive correction functions to be applied to predictions by the Świrski model. To achieve this goal, they employ synthetically generated eye images based on the LeGrand eye model (Le Grand, 1957), a widely used approximation of the physiology and optics of the human eye (see Fig. 2C). The LeGrand eye model is characterized by four physiological parameters: eyeball radius, iris radius, cornea radius, and the refractive index of the cornea. Assuming a realistic imaging setup, a large number of eye images is raytraced, varying eyeball position with respect to the camera, as well as gaze angle and pupil radius, all within physiologically plausible ranges. Subjecting the resulting images to the algorithm proposed by Świrski et al., they obtain corresponding tuples of measured and ground truth eyeball positions, gaze vectors, and pupil sizes. Multivariate polynomial regression is then employed to fit a correction function, which can be applied in realtime to similar tuples measured by means of the Świrski model on real world data.
As this method employs as many physiological parameters as the underlying LeGrand eye model, i.e. four, we refer to this approach also as 3D4p. Estimated pupil size is reported in units of physical length, more specifically [mm]. Note, this approach in particular accounts both for the effects of corneal refraction as well as changes in gaze angle.
Table 1 summarizes key properties of the three pupilsize measurement methods employed. Unless stated otherwise, in the following whenever we refer to pupil radius and corrected pupil radius, this signifies a pupil radius in [mm] as measured by the methods presented in “3D Eye model without cornea: 3D1p” and “3D Eye model with cornea: 3D4p”, respectively. Note that this is in contrast to the groundtruth pupil size, which for an in vivo scenario on independent grounds is per se unknown.
Experiments  Real world data
Groundtruth pupil size is unavailable when performing in vivo measurements. This limitation prevents obtaining a direct verification of pupillometry accuracy for any method under evaluation. Gazeangle dependency, however, provides an indirect metric for assessing the success of a given method in providing meaningful pupilsize measurements. The question we seek to answer quantitatively is therefore: how do pupilsize measurements obtained via the three methods outlined above perform in terms of this metric? To this end, we propose two protocols for in vivo measurements, both designed to reduce pupilsize fluctuations to a physiological minimum. In this way, we approximate two constant pupilsize scenarios, once producing a fairly constricted pupil, once a maximally dilated one. Given a perfect measurement method, estimated pupil size in these scenarios would be constant over time also upon changes in gaze directions, or at least only show variations which i) do not exceed those of reference periods with a static gaze and ii) do not correlate with gazeangle variation.
Bright environment
In a first series of experiments, eye videos of the left and right eye of N = 16 test subjects were recorded at 120 fps in 400x400 pixel resolution using a standard binocular Pupil Core headset (see Fig. 3), using the opensource software Pupil Capture (version 1.21.202). Of the 16 test subjects, 13 were myopic (three of them only slightly, with spherical correction <= 0.5 dioptre), the remaining three were emmetropic. Five to six recordings per subject were made, each lasting 75 s. Recordings were performed in a windowless room. Subjects were standing at a distance of about 30 cm from an evenly painted, nonglossy white wall, facing the wall which carried a small (approximately 1 cm^{2}) colored sticker as a fixation aid. The sole sources of light were two 500 W lamps on tripods, each aimed at the wall from a position approximately 1 m laterally and 1 m dorsally of the subject, thus producing a fairly even, brightly lit wall.
Prior to recording data for a given test subject, eye cameras were adjusted and the subject was instructed not to touch the headset during the recording, in order to minimize the occurrence of headset slippage.
Subjects were instructed to keep focusing on the colored sticker at all times. Each recording comprised an initial phase of 15 s of static straight ahead gaze, which provided time for the eyes to accommodate to the brightness level of the wall. This period was followed by 60 s of slow, random head rotations (referred to as sweep period). Since subjects maintained fixation on the sticker, this ensured that eye positions and rotational states with respect to the wall and thus to the diffuse but bright environment remained approximately constant, while sampling a diverse range of physiologically accessible gaze angles with respect to the headsetmounted eye cameras. In this way, we sought to create a situation in which pupilsize changes were reduced to a physiological minimum, i.e. being approximately constant with a fairly small pupil size. Recordings with each subject were performed in close succession, with no more than 60 s time in between recordings. Recordings were deliberately held short in order to avoid fatigue, a drift in pupil size, and to minimize headset slippage.
Dark environment
In a second series of experiments, eye videos of the left and right eye of the same N = 16 test subjects were recorded at 120 fps in 400x400 pixel resolution using the same binocular Pupil Core headset and Pupil Capture version. Again, five to six recordings per subject were made, each lasting 120 s. Recordings were performed in a lighttight room, in which no light sources were discernible even after 300 s of adaptation. Subjects were thus submerged in complete darkness and no fixation on any particular distance or target was possible. As compared to the bright environment, the duration of the initial static gaze period was increased in order to take the eye’s slower accommodation to darkness into account. Subjects were instructed to keep looking straight ahead at all times. Each recording comprised 60 s of static straight ahead gaze, followed by 60 s of slow, random head rotations (sweep period), while maintaining an approximately straight ahead gaze, just as in the bright environment. This way, we sought to create a situation in which pupilsize changes were reduced to a physiological minimum, this time, however, with a fairly large pupil size.
Summing bright and dark environment experiments, 233 individual recordings were made. Since always both eyes were recorded, a total of n = 2 ⋅ 233 = 466 monocular eye recordings were obtained, resulting in approximately 4 ⋅ 10^{6} individual image frames, which formed the basis of our data analysis.
Multiday recordings
To probe whether pupilradius estimates for a given subject correlate with gaze angle in a manner consistent from recording session to recording session, for two subjects experiments on three consecutive days were performed. More specifically, on each day recordings were made for each subject in the bright environment.
Experiments  Synthetic data
In order to study the gazeangle dependency of pupilsize estimates from a theoretical vantage point, we employ the raytracing pipeline presented by Dierkes et al., (2019) (see also “Gazeangle dependency of pupilsize measurements” and “3D Eye model with cornea: 3D4p”). More specifically, we generated synthetic eye images within the framework of the LeGrand eye model (Le Grand, 1957) (see Fig. 2C). During the raytracing operation, for each pixel in the image, a simulated light ray emanating from the center of an idealized pinhole camera is cast towards the eye model. Pixel color is determined by means of distinguishing four cases: (i) the simulated light ray passes the eye without intersection (resulting in a white pixel), (ii) it hits the eyeball (resulting in a beige pixel), or it hits the cornea, in which case it is refracted according to Snell’s law and continues its path until it either (iii) hits the iris (resulting in a blue pixel) or (iv) enters the pupil (resulting in a black pixel). Note that within the scope of the LeGrand eye model, cornea and aqueous humour are assumed to form a continuous medium with a uniform refractive index n_{ref}. In the generation of all synthetic eye images, we set n_{ref} = 1.3375 (Guestrin and Eizenman, 2006). Examples of generated eye images are shown in Figs. 1A and 2.
Two sets of simulations were performed, one with strictly constant pupil size and another one including pupilsize fluctuations of realistic amplitude. For the first set, in total, n = 800 simulated eye recordings were generated, mimicking our experiments in the bright and dark environment. For each simulation run, the position of the eyeball in camera coordinates was randomly chosen from a range of positions consistent with realistic setups of the Pupil Core headset. To reflect the physiological variability of human eyes, the eyeball radius, iris radius, and corneal radius of curvature were chosen randomly, each from a normal distribution with realistic mean and standard deviation (see Table 2). A constant pupil radius was randomly chosen for each simulation run from a uniform distribution between 0.5 mm and 4.5 mm, thus covering the full range of pupil radii typically encountered in humans (see e.g. Sirois and Brisson, 2014). Utilizing the above parameter values, eye images corresponding to 500 random gaze angles from a physiologically plausible range were raytraced. Images with only partially visible 2D pupil (rarely occurring for extreme eyeball positions in combination with extreme gaze angles) were discarded. The images generated in one simulation run thus correspond to an individual sweep period as recorded in our real world experiments.
For the second set of simulations, while keeping all other steps as described above, pupil size for each of the 500 eye images in a simulation run was modulated by a random factor independently drawn from a normal distribution with mean μ = 1 and a standard deviation drawn from a distribution according to the one shown in Fig. 7 (Major axis). In this way, relative pupil size over each simulated recording fluctuated with a standard deviation consistent with estimated amplitudes of pupilsize changes in our experiments (see “Sample level”). The same data processing and analysis as detailed for the real world data in the next section was applied to the synthetic data, unless stated otherwise.
Data processing & Analysis
Pupil contours were extracted from all realworld images by means of the opensource 2D pupil detector implemented in Pupil Capture (Pupil Labs, 2020a). Resulting pupil detections comprise an ellipse fit to the 2D pupil and thus, in particular, provide the length of its major axis without the need of further analysis. As the eye cameras used in the experiments showed negligible image distortion, pupil detections were only corrected for principal point offset using measured camera intrinsics.
From synthetic eye images, the 2D pupil can be trivially extracted by virtue of the corresponding pixel label obtained during raytracing. An offtheshelf algorithm was employed to obtain the bestfitting ellipse (scikitimage dev team, 2020).
In order to calculate pupil radii in the framework of either of the two modelbased methods (3D1p and 3D4p), in a first step the 3D position of the eye sphere and eyeball, respectively, needs to be estimated based on a time series of pupil detections. More specifically, we used the formulation proposed by Dierkes et al., which casts the involved minimization problem as a leastsquares intersection of lines (Dierkes et al., 2019). This algorithmic approach is implemented in the opensource pye3ddetector Python package developed by Pupil Labs (2020b). For each recording, we determined the corresponding eye sphere and eyeball position for the left and right eye postrecording, based on eyecamera frames from the sweep period. To reduce the effect of erroneous pupil detections, only the 10 % frames with highest pupildetection confidence entered the optimization. The confidence measure, a value ranging from 0.0 (lowest confidence) to 1.0 (highest confidence), was provided by the employed 2D pupil detector.
For synthetic eye images, 2D pupil segmentation is always equally confident. Thus, for each simulation run, eye sphere and eyeball position was estimated using all images generated during the respective run.
Given the estimates of eye sphere and eyeball position, uncorrected and refractioncorrected pupil radii were calculated for each eye image for both kinds of data (realworld and synthetic) as outlined in “Major axis: 2D0p”, “3D Eye model without cornea: 3D1p”, and “3D Eye model with cornea: 3D4p”. For the modelbased approaches, we employed functionality from the pye3ddetector Python package developed by Pupil Labs (2020b). The resulting time series, three per eye for each recording as well as for each simulation run, constitute our samplelevel results.
For each recording and each of the three approaches to pupilsize measurement, we determined a baseline by obtaining the median of the respective pupilsize measure over all data samples from the sweep period with a confidence of at least 0.6. Note, the experimental setup was designed to assure a pupil size that was approximately constant over time. The baseline value thus also serves as an estimate of the constant pupil size realized in each experiment. Realworld samples with a major axis deviating by more than 20 % from the corresponding baseline value were excluded from further analysis, since variations of such large amplitude could be considered erroneous pupil detections. This resulted in exclusion of less than 4 % of data samples. Analogous baselines were measured for all simulation runs. No measurements needed to be discarded, however, from the synthetic data. In order to make correlations in pupil size with gaze angle comparable between subjects and recording environments, each time series was normalized by division with the respective baseline value. We refer to the resulting values as relative pupil sizes.
We ultimately seek to shed light on the variation of pupil size as a function of gaze angle. Both modelbased methods, for each eye image also provide a gazeangle estimate. The respective estimates, however, depend both on model assumptions as well as the choice of modelspecific parameters. Striving for a more objective measure, we decided to use circularity C of the 2D pupil ellipse instead. More specifically, circularity is defined as the ratio of minor and major axis of the 2D pupil ellipse. Circularity values thus range from 0 to 1, with 1 corresponding to a perfectly circular pupil image (see Fig. 4). The latter case is observed when looking straight at the camera, with the eyeball being positioned on the camera’s optical axis (see Fig. 1A, left synthetic eye image). With increasing angle between the optical axis of the eye and the camera, respectively, circularity of the 2D pupil ellipse decreases (see Fig. 1A, right synthetic eye image). For offaxis eyeball positions, i.e. when rotational symmetry is broken, circularity can differ slightly for identical gaze angles. Since on theoretical grounds these deviations are expected to be small, however, we decided to use circularity of the 2D pupil ellipse as a convenient proxy for gaze angle. In particular, this strategy allowed us to average pupilsize data from different but equivalent gaze directions.
More specifically, for each subject, each eye, and each approach to pupilsize measurement, we performed the following data aggregation. Relative pupil sizes as obtained during the sweep period were binned according to circularity into ten bins of equal width, spanning the theoretical range from 0 and 1. The weighted mean over all recordings, both from the bright as well as the dark environment, was then calculated in each bin. As weight for each sample we used the corresponding pupildetection confidence.
In the case of the synthetic data, each pupilsize sample contributed equally. To further make results comparable between subjects as well as simulation runs, the resulting values were normalized by dividing by the value of the bin corresponding to the highest circularity. On the subject level, per eye all recorded data was thus aggregated into normalized average relative pupil size as a function of circularity.
At the population level, in each bin we further calculated the mean and standard deviation of all curves obtained on the subject level.
For ease of reference, below we summarize the data processing steps in condensed form:

1.
Sample level

i)
Detect pupil ellipses in all recorded eye images.

ii)
Estimate 3D eye sphere and eyeball position based on highconfidence pupil detections from sweep period of each recording.

iii)
Determine pupil size for each eye image.

iv)
Calculate baseline, i.e. median of pupilsize estimates during sweep period with pupildetection confidence larger than 0.6.

i)

2.
Subject level

i)
Filter out all data samples with a pupil ellipse with major axis deviating from the corresponding baseline by more than 20 %.

ii)
For each eye, calculate the confidenceweighted mean of pupil sizes divided by baseline in ten circularity bins between 0 and 1. The mean is taken over all recordings for a given subject.

iii)
Normalize by average pupilsize value of the bin corresponding to the highest circularity, i.e. C between 0.9 and 1.0.

i)

3.
Population level

i)
Calculate mean and standard deviation over test subjects and left and right eyes within each bin.

i)
Results
Sample level
Sample eye images recorded for two subjects in the dark and bright environment, respectively, are displayed in Fig. 4. Pupil ellipses as detected by Pupil Capture are shown (pink lines) with their circularity being indicated. As can be seen from the images, rotation of the eye with respect to the camera induces changes in the circularity of the observed pupil ellipse. In this study, we use circularity as a proxy for gaze direction, as motivated in the previous section. In particular, note the larger pupil dilation in the dark compared to the bright environment.
Time series of pupilsize measurements obtained via all three methods are shown in Fig. 5 (blue, green, and red lines). More specifically, data from a typical monocular recording obtained in the dark and bright environment is presented in the left and right column, respectively. Note the different extent of the vertical axes used to present the data from the dark versus the bright environment. Also shown are corresponding circularity values (orange lines). Vertical dashed lines at time t = 0 s represent the transition from static straight ahead gaze to the sweep period.
In the sweep period (t > 0 s), oscillatory variations in circularity are apparent in both recordings. These variations provide evidence that the performed random head rotations were efficacious in inducing gazeangle changes relative to the recording camera. A correlating variation can be discerned in the results of all three measurement methods, showing that to a certain extent all methods exhibit gazeangle dependency on the sample level.
In order to further probe the observed correlation, in Fig. 6 we show in a single panel results from all three methods normalized to their respective baseline. More specifically, the data shown corresponds to the period 25 s < t < 40 s in the dark environment example shown in Fig. 5. Several observations can be made. While the major axis correlates positively with circularity, both modelbased methods correlate negatively with it. Upon a rotation away from the camera, the major axis thus tends to underestimate pupil size relative to the value at C = 1, while both modelbased approaches tend to overestimate it. As to their amplitude, major axis and 3D1p exhibit variations of similar size. The amplitude of variation in the 3D4p results in comparison appears reduced. While this reduction is indicative of the partial success of the employed refractioncorrection scheme, it is not able to fully eliminate all gazeangle dependency on the sample level.
In addition to gazeangle induced changes of measured pupil size, the example time series shown also exhibit pupilsize fluctuations on longer time scales. This becomes apparent in particular in the bright environment example, in which pupil size as measured by all three methods increases by about 10 % over the first 30 s shown. While not as obvious in the dark environment example, there is reason to suspect pupilsize fluctuations also in this case. Lacking a visible fixation target, in the dark environment the accommodative state of the eye is ill defined. In particular, we cannot exclude that the Pupil Near Response (PNR)  which entails pupilsize changes  was triggered at random points in time. In addition, it has been reported that in darkness and/or in the absence of direct external stimuli, the size of the pupil can fluctuate with frequencies around 0.51.0 Hz and amplitudes of the order of a few percent (Köles, 2017; Sirois and Brisson, 2014; Mathôt et al., 2018). In order to estimate the extent to which pupil size was fluctuating over the time course of our experiments, for all recordings and all three measurement methods, we calculated the standard deviation of relative pupil size over a time window of 10 s prior to the sweep period (see Fig. 7). Participants were instructed to hold a static gaze prior to the sweep period. These measurements were therefore not influenced by PFE. As the resulting histograms show, fluctuations relative to the baseline for all three approaches had standard deviations of up to 10%, with a mean of approximately 5%. Thus, while our experiments were carefully designed to provide controlled light conditions, due to physiological factors outside of our control, a constant pupil size in any given recording could only be approximately realized.
Subject level
Physiologically induced pupilsize fluctuations bear no correlation with gaze direction. Aggregating pupilsize measurements in terms of pupilellipse circularity allows for integrating data from different but equivalent gaze directions as well as independent points in time. It thus provides a tool for averaging out pupilsize fluctuations that are uncorrelated with gaze direction. Note, we will further analyze the effect of fluctuating pupil size in “Population level” by means of our simulations of synthetic eye recordings.
In Fig. 8, for two subjects we present pupilsize data aggregated at the subject level. More specifically, we show normalized relative pupil size as a function of circularity (see “Data processing & Analysis”). For each subject, data from the left eye is displayed, with solid, dotted, and dasheddotted lines corresponding to recording sessions performed in the bright environment on three consecutive days (see “Multiday recordings”). Note, in the bin corresponding to the largest circularity (0.9 < C < 1.0), all curves coincide due to normalization. The proposed statistics thus reveals by what fraction pupil size is underestimated/overestimated upon a rotation of the eye away from the recording camera; with a circularity of C = 0.4 corresponding to a gaze angle of about 50^{∘} with respect to the camera’s optical axis.
While the major axis (red lines) for both subjects tends to progressively underestimate pupil size with decreasing circularity (up to 7.5% at C = 0.4), the pupil radius obtained by the 3D1p approach (blue lines) tends to increasingly overestimate it. Note, this is in line with the samplelevel results presented in the last section (see Fig. 6). In case of the 3D1p approach, however, the extent of overestimation differs between subjects (about 7.5% for subject A at C = 0.4, about 3% for subject B at C = 0.4). As for the results obtained by the 3D4p approach (green lines), for subject A correlation with circularity is negative, for subject B it is positive. More importantly, while overestimation of pupil size for subject A is as low as about 2.5% over the range of circularities shown, for subject B an underestimation of about 4% at C = 0.4 can be discerned. In other words, while refraction correction renders measured pupil size for subject A largely gazeangle independent, for subject B it tends to overcorrect.
Note, intersubject differences between curves obtained with the same approach are more pronounced than intrasubject differences for corresponding curves recorded on different days. Our results thus strongly suggest, while daytoday variations in hardware setup (camera adjustment, pose of the eye tracker on the head of the subject, etc.) modulate results slightly, the observed gazeangle dependency is predominantly shaped by subjectspecific factors such as eye physiology.
Population level
As shown in the last two sections, on the sample and subject level none of the three measurement methods guarantees gazeangle independent measurements of pupil size. In practice, however, pupillometry experiments are most often concerned with measuring pupil size as averaged over a large number of test subjects and repetitions of stimulus presentation, i.e. with gauging pupillary responses at the population level. We therefore continue with a presentation of our results at this level of data aggregation.
To this end, in Fig. 9A for all subjects, we show subjectlevel curves of normalized average relative pupil size as a function of circularity. Note, since both the left and right eye was recorded, each subject contributes two curves. In line with results discussed in previous sections, this data shows that the major axis tends to underestimate pupil size with decreasing circularity, while 3D1p tends to overestimate it. The refractioncorrected 3D4p approach exhibits over/underestimation, depending on subject and eye.
Mean curves as calculated per circularity bin are shown for each measurement method in Fig. 9B. Standard deviations around the mean are indicated by shaded regions. While the major axis and 3D1p show deviations of more than 5 % at C = 0.4 (albeit with opposite sign), the results obtained with 3D4p stay within ± 1 % (dashed horizontal lines) of the reference value at largest circularity. These results clearly demonstrate that the refractionaware approach by Dierkes et al., here referred to as 3D4p, is successful in providing pupilsize measurements which are independent of gaze angle at the population level. In contrast, the major axis and the approach by Świrksi et al., here referred to as 3D1p, show a similar level of gazeangle dependency.
In Fig. 9C, D (no pupilsize noise) and Fig. 9E, and F (with pupilsize noise), we present analogous data aggregations based on simulations performed with our raytracing system (see “Experiments  Synthetic data”). We deal with the noiseless case first. In particular, each curve in Fig. 9C reflects a random choice of plausible parameters characterizing eyeball physiology and setup of the eye tracker. These results show that at the population level, the extent of gazeangle dependency observed in our realworld measurements is in quantitative accord with theoretical predictions for all three measurement methods. Our simulation results also give an indication as to the expected intersubject variability of gazeangle dependency. We find that standard deviations measured in our realworld experiments are larger than in simulations (cf. Figure 9B and D), i.e. only part of the observed variability can be explained by reference to variation in physiological eye parameters.
When incorporating pupilsize noise into our simulations, population averages are largely unchanged (cf. Fig. 9D and F). This observation provides evidence that our aggregation scheme is indeed successful in averaging out the effect of unavoidable pupilsize fluctuations (see Fig. 7). At the same time, subject level curves clearly show fluctuations more similar to our realworld data (cf. Fig. 9A and E). Overall, the observed standard deviations, while slightly larger when adding noise in comparison to the noiseless case, are still lower than those of the realworld data (cf. Fig. 9B, D, and F).
We speculate that the remaining variation can be accounted for by measurement noise in the realworld data, resulting inaccuracies in the estimates of eye sphere and eyeball position, and physiological variation in actual eyes that is not captured by the LeGrand eye model.
Discussion
In this work, we have investigated the extent of PFEinduced gazeangle dependency in three methods for pupilsize measurement in the context of headmounted eye tracking. Next to an imageimmanent method (major axis), we have considered two modelbased approaches (3D1p, 3D4p). Presenting both experimental realworld data as well as results from a simulation study, we have analyzed the correlation of pupilsize estimates with gaze angle at the sample, subject, and population level.
Our two experimental setups were carefully designed to minimize pupilsize changes occurring over the time course of each recording, i.e. to approximate constant pupilsize scenarios. This did not, however, prevent physiologically unavoidable pupilsize fluctuations with an amplitude comparable to the effects to be measured. Such pupilsize changes constitute a potentially confounding factor for the assessment of PFEinduced gazeangle dependency.
A viable strategy for reducing pupilsize fluctuations even further would be the application of a mydriatic drug such as Cyclopentolate that produces sustained pupil dilation. Such an approach, however, constitutes an invasive procedure requiring medical supervision and would furthermore limit experiments to large pupil sizes.
Our analysis instead hinges on the extensive averaging of pupilsize data in terms of 2D pupil circularity, both at the subject as well as the population level. At the resolution of the employed neareye camera, 50 pixels is a common major axis length of the corresponding 2D pupil in the bright environment (see Fig. 5). A 2 % change in pupil size  a relevant effect in the domain of pupillometry for cognitive sciences  thus corresponds to merely 1 pixel. The proposed averaging strategy is well suited for uncovering such small but systematic dependencies, as it not only combines data from different time points but also from different but equivalent gaze directions. In particular, since pupilsize fluctuations under even illumination are independent of gaze direction, our approach largely eliminates the potentially confounding effect of nonconstant pupil sizes. We have further validated the robustness of our approach by incorporating pupilsize fluctuations of realistic amplitude in simulations of synthetic eye images mimicking the performed realworld experiments.
Of all three measurement methods, only the refractionaware 3D4p approach resulted in pupilsize estimates that were gazeangle independent at the population level. Both other approaches were found to exhibit systematic errors at this level of data aggregation. More specifically, with decreasing circularity, pupil sizes tend to be either underestimated (major axis) or overestimated (3D1p).
Our strategy of aggregating pupil size as a function of circularity suggests an approach for developing a posthoc correction scheme. By fitting the observed populationlevel dependency and applying a corresponding multiplicative correction to measured pupil radii in a circularitydependent manner, the gazeangle dependency of population averages for the major axis and 3D1p approach potentially could be reduced.
Even assuming the viability of such posthoc correction, however, the 3D4p approach provides advantages. Most importantly, it is not necessitating any prior measurements mapping out systematic errors in pupilsize estimates as a function of circularity. Both modelbased approaches are superior to the major axis in that they provide estimates of the actual physical dimensions of the ocular aperture stop in [mm]. At the same time, they also provide estimates of eyeball position and gaze direction, thus integrating pupilsize estimation into a system of broader scope. Since the 3D4p approach is expected to provide more accurate results in absolute terms than the 3D1p approach (Dierkes et al., 2019), we overall believe it to be the most convincing choice for many use cases.
Within the cognitive sciences, it is common practice to average pupillometric results at the population level to arrive at statistically meaningful results, due to the small magnitude of effects to be measured, the differences in individual eye physiology as well as reaction to stimuli, and the noisy nature of the underlying video or image data. Our work provides evidence as to the effectiveness of such averaging strategy in the case of headmounted eye trackers and gives a quantitative estimate of expected systematic errors depending on the measurement approach chosen.
At the sample and subject level, in contrast, all three methods exhibit gazeangle dependency to some extent. Subjectlevel curves are spread around corresponding population averages, with the observed spread increasing with decreasing circularity. The observed spread relative to the observed systematic error at population level is similar for all approaches. In particular, while largely being free of systematic errors at the population level, also the 3D4p approach fails to provide PFEfree pupilsize measurements at the subject level.
Our data shows that gazeangle dependency on the subject level is largely consistent between recordings obtained on different days, thus strongly suggesting that it is individual variations in eye physiology shaping the observed extent of PFE in each subject. All three approaches are ultimately based on an analysis of the 2D pupil contour. Given a 3D pupil, the exact shape of the 2D pupil image depends on personspecific parameters, e.g. corneal radius, which are determining the optical characteristics of the eye. It thus stands to reason that any approach aiming at PFEfree pupilsize measurement on the subject level, ultimately needs to account for individual differences in eye physiology.
A conceivable posthoc correction approach would be to map out the dependence of pupil size on circularity for a given subject in a set of experiments such as the ones performed in this study. Using the observed relation as de facto calibration, subjectlevel results could be corrected by applying a fitted multiplicative factor in a circularitydependant manner.
Note, the current implementation of the 3D4p approach assumes average human eye parameters. It is thus plausible that personspecific physiological deviations from those averages  together with particularities in the eye physiology which are not modelled at all  will lead to imperfect correction of pupilsize measurements on the subject level. A potentially less timeintensive and more principled approach would thus be to make personspecific parameters in the 3D4p approach adjustable, in order to account for the specifics of a subject’s eye physiology. While raytracing synthetic images for a given parameter set and deriving corresponding refraction functions is feasible, it remains an open question, however, whether relevant ocular parameters could be measured easily with sufficient accuracy in order for this approach to be practical.
Also note, the LeGrand eye model constitutes an approximation of ocular optics, as it does not take into account e.g. the effect of nonsphericity of the cornea, variations in corneal thickness, noncircularity of the pupil, or asymmetries in eyeball shape. Facilitating pupilsize measurement which is truly gazeangle independent on the subject level might thus also necessitate the use of more realistic eye models, such as the Navarro eye model, albeit at the cost of an increased number of personspecific parameters to be determined. Since raytracing is also feasible for eye models of increased veracity, we believe, however, the general approach of determining refractioncorrection functions in a personspecific manner by means of synthetic eye images to be a promising strategy.
Notes
The image of the pupil is in fact not an exact ellipse, as e.g. pointed out in Fedtke et al., (2010): “[...] the shape of the peripheral entrance pupil does not correspond to an ellipse as often assumed. Instead, although mathematically different, it resembles the shape of a convex https://en.wikipedia.org/wiki/Limacon”. In the following, however, we will disregard this technicality and refer to the pupil image as being an ellipse.
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Both authors are employees of Pupil Labs, Berlin, Germany. No funds, grants, or other support was received. All test subjects were employees of Pupil Labs and consented to partake in the study. None of the experiments was preregistered. The data and materials as well as example code which implements all data analysis steps described herein can be accessed under (Pupil Labs, 2021). Open source packages used for data analysis are referenced.
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Petersch, B., Dierkes, K. Gazeangle dependency of pupilsize measurements in headmounted eye tracking. Behav Res 54, 763–779 (2022). https://doi.org/10.3758/s13428021016578
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DOI: https://doi.org/10.3758/s13428021016578
Keywords
 Pupillometry
 Pupil foreshortening error
 PFE
 Eye tracking
 3D eye model
 Corneal refraction