11affiliationtext: Institute for Ophthalmic Research, University of Tuebingen,
Elfriede-Aulhorn-Straße 7, Tübingen, 72072 22affiliationtext: School of Social
Science and Psychology, Western Sydney University, New South Wales, Australia
33affiliationtext: Carl Zeiss Vision International GmbH, Turnstraße 27, 73430
Aalen

#  Virtual Reality (VR)as a testing bench for consumer optical solutions: A
machine learning approach (GBR) to visual comfort under simulated progressive
addition lenses (PALs)distortions

Miguel García García  Yannick Sauer  Tamara Watson  Siegfried Wahl

###### Abstract

For decades, manufacturers have attempted to reduce or eliminate the optical
aberrations that appear on the progressive addition lens’ surfaces during
manufacturing. Besides every effort made, some of these distortions are
inevitable given how lenses are fabricated, where in fact, astigmatism appears
on the surface and cannot be entirely removed or where non-uniform
magnification becomes inherent to the power change across the lens. Some
presbyopes may refer to certain discomfort when wearing these lenses for the
first time, and a subset of them might never adapt. Developing, prototyping,
testing and purveying those lenses into the market come at a cost, which is
usually reflected in the retail price. This study aims to test the feasibility
of virtual reality for testing customers’ satisfaction with these lenses, even
before getting them onto production. VR offers a controlled environment where
different parameters affecting progressive lens comforts, such as distortions,
image displacement or optical blurring, can be analysed separately. In this
study, the focus was set on the distortions and image displacement, not taking
blur into account. Behavioural changes (head and eye movements) were recorded
using the built-in eye tracker. Participants were significantly more
displeased in the presence of highly distorted lens simulations. In addition,
a gradient boosting regressor was fitted to the data, so predictors of
discomfort could be unveiled, and ratings could be predicted without
performing additional measurements.

Keywords: virtual reality distortions comfort progressive addition lenses eye-
tracking

MGG Orcid 0000-0001-7379-0080
YS Orcid 0000-0002-7513-341X
TW Orcid 0000-0001-7899-0858
SW Orcid 0000-0003-3437-6711
Corresponding author: Miguel García García
<EMAIL_ADDRESS>
##  1 Introduction

Progressive addition lenses (PALs)provide presbyopes with clear vision at
different distances, with the use of a single ophthalmic lens (Poullain and
Cornet, 1911). Firstly patented in 1907 (Aves, 1907), multifocal ophthalmic
lenses have undergone a great deal of evolution (Sullivan and Fowler, 1988).
Although their introduction to the market was neither rapid nor widely
accepted, being only in 1959, when precursors of modern progressive lenses
became commercially available (Maitenaz, 1969; Volk and Weinberg, 1962), over
the years, progressive lenses have become the most popular solution for
presbyopia.

In general, the fabrication of progressive power lenses leaves inevitably
residual surface astigmatism that leads to peripheral distortions (Minkwitz,
1963; Esser et al., 2017). How this residual astigmatism is spread across the
surface was usually employed to classify PALs into soft (more spread) and hard
designs (more concentrated) (Atchison, 1987). However, nowadays, these lenses
navigate between both classes, diffusing the borders as to whether they belong
to one or another category, with manufacturers aiming to obtain the best
combination of parameters for each group of individuals.

Besides their popularity and the fact that they do seem to improve wearers’
quality of life (Ahmad Najmee et al., 2017), not everyone gets used to them.
Scientific literature is scarce when it comes to gathering knowledge about why
few "rookie" wearers suffer in adapting to these lenses or which could be the
main factors contributing to this. Some papers mainly refer to spectacles in
general and report errors in the prescription as the primary source of
discomfort (Bist et al., 2021). However, assuming no errors from the optician,
a small group still reports discomfort due to distortions and multifocality.
Alvarez et al. (2017) proposed that non-adopters may have a weaker ability to
modify their convergence and a reduced rate and magnitude of phoria
adaptation. Yet, individuals who reject these lenses report blurred vision,
headaches, perceived movement of the peripheral visual field (a.k.a. swim),
balance issues and even nausea (Cho et al., 1991; Han et al., 2003), symptoms
more likely associated with poor adaptation to novel visual conditions
(Alvarez et al., 2009).

Traditionally, as reported by Ogle and Saunders W.B. (1950), one of the most
significant handicaps when analysing the tolerance to distortions induced by
ophthalmic lenses was the difficulty in separating them from blur (Barbero and
González, 2020). With the computational development of the last decades, it is
entirely possible to present an image that is solely distorted (Habtegiorgis
et al., 2017; Sauer et al., 2020) or blurred (Sawides et al., 2010; Vinas et
al., 2012), helping to understand how our visual system adapts to these
changes. However, these experiments are somewhat forced, built upon certain
constraints, such as specific gazing or a fixed head position, and while they
do drive knowledge forward, they often lose sight of the big picture.

On the other hand, one can analyse eye-tracking data while wearing PALs
(Hutchings et al., 2007), bringing more natural conditions at the expense of
not fully comprehending which features are affecting what since they can not
be separated. In addition, the potential source of error that arises when
fitting PALs cannot be ignored. All in all, virtual reality offers a perfect
test bench as it can replicate the effect of distortions and blurring apiece
while maintaining the freedom of movement and immersion of virtual
environments.

This study aims to test the feasibility of virtual reality to assess,
benchmark and rate the discomfort that four different ophthalmic lenses with
various refractive powers can introduce, given the distortions they present.
Subjective grading was performed continuously after finishing a simple task
involving navigation, locomotion, scene exploration, and grasping to recognise
the subjects’ ability to perceive such simulated distortions and assess their
comfort when affected by them. Possible behavioural changes prompted by
optical distortions were also sought, and a model was developed to predict
what the discomfort might be like without requiring additional measurements.

##  2 Material & Methods

###  2.1 Subjects and informed consent

A total of 18 naïve subjects (9 males / 9 females) participated in the course
of the study. The participants were aged between 19 and 30 years (mean = 24;
SD = 3). None of the subjects presented a prior history of problems using a
virtual reality headset or motion issues, nor did they present any refractive
error that could have compromised their vision during the experiment or
influenced their perception due to the usual wear of their lenses.

###  2.2 Ethics

The study adhered to the tenets of the Helsinki Declaration (2013) and
subsequent amends. The ethics authorisation to perform the measurements was
granted by the Ethics Committee at the Medical Faculty of the Eberhard-Karls
University and the University Hospital Tübingen with the ID 986/2020BO2.
Before data collection, the experiment was explained in detail to the
participants, and written informed consent from each participant was stored.
All data was pseudo-anonymised and stored in full compliance with the
principles of the Data Protection Act GDPR 2016/679 of the European Union (The
European Parliament and the Council of the European Union, 2016).

###  2.3 Set-Up

A cuboidal booth of $$3\text{\,}$\times$3\text{\,}\mathrm{m}$$ defined the
physical space around which the participants were entitled to freely move. On
every upper corner (4 in total), one HTC Vive lighthouse base station tracker
v2 (HTC Cooperation, Xindian, Taiwan) recorded the position and orientation of
the headset as well as the controller used. The StarVR One (StarVR, Taipei,
Taiwan) head-mounted display was used to present the virtual environment to
the participants. This head-mounted display (HMD)has an effective field of
view (FoV)of $182\text{\,}\mathrm{\SIUnitSymbolDegree}$ by
$99\text{\,}\mathrm{\SIUnitSymbolDegree}$ (Sauer et al., 2022), and an average
eye relief (or distance from lens/display to the eye of
$18.2\text{\,}\mathrm{mm}$ (15.8 to $23.9\text{\,}\mathrm{mm}$). The display
has a resolution of
$$2240\text{\,}\mathrm{px}$\times$1792\text{\,}\mathrm{px}$$, a maximum
luminance of $68.4\text{\,}\mathrm{cd}\text{\,}{\mathrm{m}}^{-2}$ and the
refresh rate fixed at $90\text{\,}\mathrm{Hz}$. This device presents eye-
tracking capabilities through Tobii Pro libraries. For reference, the display
area covered by the eye-tracker, as reported by the manufacturer, is plotted
in Figure 1. To provide input and locate the hand position, a standard HTC
Vive controller was used.

Figure 1: Area of display resolution of the VR headset(gray) as well as area
in pixels covered by the eye tracker(orange) as provided by the manufacturer.

####  2.3.1 Virtual environment & Computer

The virtual environment was constructed using the rendering engine Unity
version 2019.4.25.f1 (Unity Technologies, California, USA) and Blender 3.0
(Stichting Blender Foundation, Amsterdam, Netherlands). The experiment was
written in C# and the eye-tracking data was recorded using the Tobii Pro
(Tobii Pro AB, Danderyd, Sweden) eye-tracking libraries. A view of the virtual
reality environment used is depicted in Fig 2.

Figure 2: View of the virtual environment. The blue square denotes where the
instructions were located in the virtual environment, and the rating user
interface (UI)was presented. The red circle is located where the cubes need to
be placed. Finally, the green star is placed upon the cube spawns’ origin.

The whole experiment ran on a Windows 10(20H2) PC with an Intel(R) Core
i7-10700, 16 GB of RAM and an Nvidia 3080 GPU.

###  2.4 The task

After providing informed consent and having the experiment explained in
detail, participants had time to ask questions. They then wore the virtual
reality headset to acclimate to the scene by wandering around the virtual
environment (a replica of a lab room) before the actual assignment started.
Across the task, participants were requested to position three different
virtual cubes from point A (Green star) to point B (Red circle)(See Fig 2).
The location at which those cubes spawned was randomised between several pre-
defined positions in the shelf unit. The cubes’ colours, the patterns they
presented on their surface and the location where they were supposed to be
placed were also randomly changed across trials. Instructions on where the
cubes had to be placed were provided on a virtual whiteboard (Blue square)
opposite the cubes’ spawn origin. Thus, participants were forced to move and
look around in the virtual environment during each trial.

####  2.4.1 Ratings of discomfort

After every cube was correctly placed on the plate, a UI panel with a slider
popped up in front of the participant. In this panel, the participants were
asked to report their level of visual discomfort on a scale from 0 (no
discomfort) to 7 (highest discomfort). Previous to providing this feedback,
participants were instructed to rate only based on how the distortions
affected their perception of their visual comfort and to not take into account
factors such as possible frame drops or the image resolution. Given the rating
and confirmation, the trial was completed, and a new trial started.

A total of 25 trials were answered (five per lens condition), with an average
duration of a trial of $36\text{\,}\mathrm{s}$ (SD = $12\text{\,}\mathrm{s}$,
range from $18\text{\,}\mathrm{s}$ to $101\text{\,}\mathrm{s}$).

###  2.5 Lens Conditions - Distortion simulation

Four progressive additional lenses of the same type, but different
prescription power were used. Different addition powers or spherical power
prescriptions do modify how distortions are present. The distortions profiles
were provided by the manufacturer from a set of lenses with the following
parameters fixed: a refractive index of 1.5, a pantoscopic angle of
$7.5\text{\,}\mathrm{\SIUnitSymbolDegree}$, a $5.28\text{\,}\mathrm{mm}$ base
curvature, a back vertex distance of $12\text{\,}\mathrm{mm}$ and a corridor
length of $14\text{\,}\mathrm{mm}$. Two of these lenses had non spherical
power and addition powers from +2 and $+3\text{\,}\mathrm{D}$, and the other
two lenses had an addition power $+2\text{\,}\mathrm{D}$ and a spherical
refractive correction of +2 and $-2\text{\,}\mathrm{D}$. In line with modern
PALs, these lenses have their "umbilical line" tilted, with the near point
shifted towards the nasal side to account for vergence.

A set of grid points ($x$, $y$) on a plane perpendicular to a straight gaze
direction is perceived in a distorted position ($x_{d}$, $y_{d}$) when seen
through a progressive addition lens. The displacement between ($x_{d}$,
$y_{d}$) and ($x$, $y$) was pre-computed for each lens using ray tracing based
on these lenses’ digital lens surface data. The displacement in the horizontal
and vertical coordinates were encoded into a raster image (EXR format) where
the red colour channel was used for the shift of the pixel in the horizontal
direction and the green colour for the vertical component.

These distortions were then applied pixel-wise to the image displayed in the
virtual reality headset through a custom shader in Unity 3D, written in HLSL.
The amount of pixel displacement that these lens simulations prompted in
visual angle, along with the magnification, skew, aspect or rotation, as seen
by the participants, can be observed in Fig 3.

Figure 3: Representation of how distortions affect a grid and the image
observed (background), and maps of pixel displacement in visual angles as
observed from the subject’s perspective for the OD, as well as magnification,
differences in aspect ratio, skew and rotation calculated from the first
derivatives of the distortions.

The angular displacement in the visual field as observed was computed using
the following formula:

$\displaystyle\begin{split}\mathrm{Displacement\hskip
2.84544pt(V_{(Visual\hskip 2.84544ptangle\hskip 2.84544ptin\hskip
2.84544ptdegrees)})}=\\\ cos^{-1}\left(\frac{x_{d}\times x+y_{d}\times
y+1}{\sqrt{(x_{d}^{2}+y_{d}^{2}+1)\times(x^{2}+y^{2}+1)}}\right);\end{split}$
(1)

Where ($x$, $y$) are the coordinates of the original grid points and ($x_{d}$,
$y_{d}$) are the location of those points after the distortion is applied.

###  2.6 Tracking

On every frame, the position coordinates of the head-mounted display
(HMD)along with the rotation quaternions and gaze vectors were stored.

####  2.6.1 Head-tracking

Head rotation values were converted using _quat2eul_ into Euler angles
following the MATLAB coordinates system for analysis and plotting. The table
2.6.1 summarises the different coordinates systems that are used in the main
libraries of this study.

vspan=minimal  Program Library Coordinate System X Y Z Euler angles order
Unity 3D Left-handed $\rightarrow$ $\uparrow$ $\nearrow$ z,x,y MATLAB Right-
handed $\nearrow$ $\leftarrow$ $\uparrow$ z,y,x ,

Table 1: Coordinates system references for each software system. $\rightarrow$
\- right; $\leftarrow$ \- left; $\uparrow$ \- up; $\nearrow$ \- forward.

On every trial the yaw, pitch and roll were computed using the HMD quaternions
as described in the equation 2.

$\displaystyle\begin{split}\mathrm{Tait\hskip 2.84544ptBryan\hskip
2.84544ptAngles\hskip 2.84544pt(roll,pitch,yaw)\hskip 2.84544pt}=\\\
\textit{quat2eul}(rotW,rotZ,-rotX,rotY,XYZ);\end{split}$ (2)

The roll data (tilt of the head) was further characterised with a Von Misses
distribution fit (See equation LABEL:eq:vonmis).

$\displaystyle\begin{split}\mathcal{M}\hskip 2.84544pt(\theta|\hskip
2.84544pt\mu,\kappa)={\left[2\pi I_{0}(\kappa)\right]}^{-1}exp\\{\kappa\hskip
2.84544ptcos(\theta-\mu)\\};\\\ \end{split}$ (3)

Where $\mu$ is the mean direction, $I_{0}$ is the bessel function of the first
kind, $\kappa$ is the concentration parameter, and $\theta$ refers to the
angle in radians from $-\pi$ to $\pi$.

The pitch (or head inclination up to down) was described using a combination
of two Von Misses distributions. And finally, the yaw (or horizontal head
rotation) was represented by the fit (See equation LABEL:eq:pb) of two inverse
Power Batschelet distributions (Mulder, 2019; Mulder et al., 2020), as the
data was highly peaked towards the cube spawns origin and the horizontal
location of the plate, whiteboard and rating UI panel.

$\displaystyle\begin{split}f_{\mathcal{PB}}\hskip 2.84544pt(\theta|\hskip
2.84544pt\mu,\kappa,\lambda)=\left[\mathcal{K}_{\kappa,\lambda}^{\ast}\right]^{-1}exp\\{\kappa\hskip
2.84544ptcos\hskip 2.84544ptt_{\lambda}^{\ast}(\theta-\mu)\\};\\\
\mathcal{K}_{\kappa,\lambda}^{\ast}=\int_{-\pi}^{\pi}exp\\{\kappa\hskip
2.84544ptcos\hskip 2.84544ptt_{\lambda}^{\ast}(\theta-\mu)\\}\hskip
2.84544ptd\theta;\\\ t_{\lambda}^{\ast}\hskip
2.84544pt(\theta)=sign(\theta)\pi\left(\frac{|\theta|}{\pi}\right)^{\gamma(\lambda)};\\\
\gamma(\lambda)=\frac{1-c\lambda}{1+c\lambda};\\\ \end{split}$ (4)

where c = 0.04082284 as noted by Mulder (2019).

####  2.6.2 Eye Tracking

The eye-tracking data was obtained using the Tobii Pro SDK 1.10 for Unity. The
SDK was modified to further record the hit-point of the gaze vector in the
world coordinates.

##### Fixations

While recording the eye-tracking data, on every frame, the combined (head and
gaze) origin and direction were used to cast a ray within Unity until it hit a
collider in the scene. The coordinates of this hit-point (x,y,z) were later
used to detect fixations. If a cluster of continuous hit-points stood within
the volume of a sphere of $0.05\text{\,}\mathrm{m}$ radius, during more than
$200\text{\,}\mathrm{ms}$ (See van der Lans et al. (2011) for a review on
average fixations duration while visual search and scene viewing), it was
considered a fixation. The amounts of fixations performed per minute and their
average duration were recorded on every trial.

##### Saccades

To estimate saccadic eye movements, the gaze ray from the right eye (in
headset relative coordinates) was transformed into polar ($\theta$) and
azimuthal angles ($\phi$). The median angular distance travelled ($\psi$)
between the original frame vector, and the five subsequent frames was computed
following the equation 5. Then, the angular speed was estimated as the angular
distance per second, on which a Savitzky-Golay filter (Dai et al., 2016) was
applied.

$\displaystyle\begin{split}\phi=\arctan\left(\frac{y}{x}\right);\\\
\theta=\arccos(z);\\\ \psi=\\\
\arccos\Bigl{(}(\sin(\theta_{n+i})\times\sin(\theta_{n})\times\cos(\phi_{n+i}-\phi_{n})\\\
+\cos(\theta_{n})\times\cos(\theta_{n+i})\Bigr{)};\\\ \end{split}$ (5)

A speed threshold of
$50\text{\,}\mathrm{\SIUnitSymbolDegree}\text{\,}{\mathrm{s}}^{-1}$ and a
minimum distance ($\psi$) of $1\text{\,}\mathrm{\SIUnitSymbolDegree}$ were
defined as thresholds for defining a saccade. The number of saccades per
minute, the average amplitude and maximum peak velocity performed on every
trial were recorded.

##### Eye tracking to measure average observed displacement, magnification,
aspect, skew and rotation

Every gaze position was projected over the displacement, magnification, skew,
rotation, and differences on aspect ratio maps, and the observed absolute and
relative ("foveally") pixel displacement, magnification, skew, rotation and
aspect ratio parameters were computed by summing the values at gaze position
over all the frames of the trial. The mean value and standard deviation per
trial were recorded and compared.

##  3 Analysis & Preprocessing of the Data

All the statistical analysis and plotting were performed in MATLAB 2020b
(Mathworks Inc, Natick, CA, USA) and Python 3.10 with sci-kit learn 1.0.2
(Pedregosa et al., 2011).

##  4 Results

###  4.1 Visual discomfort

On every trial, the participants rated their visual discomfort on a scale from
0 to 7, with zero being normal and seven being the most uncomfortable.
Discomfort is, in fact, a subjective parameter that is highly susceptible to
vary between subjects and depends on where the participant sets the threshold
towards unpleasantness. Given the same distortion and conditions, two subjects
can be uncomfortable or not. To standardise our answers and limit the
subject’s susceptibility, the ratings were normalised using the ratings where
no lens distortion was presented as a baseline. The formula used can be
observed in the equation 6.

$\displaystyle\begin{split}R_{\\{i\\}}\hskip 2.84544pt=R_{\\{i\\}}\hskip
2.84544ptorig\times\frac{10}{7}+1;\\\ R_{\\{i\\}}\hskip
2.84544ptnorm=\frac{R_{\\{i\\}}}{\overline{R}_{\\{baseline\\}}};\end{split}$
(6)

The distribution of visual discomfort ratings was not parametric in any of the
conditions tested. Thus, a non-parametric test (Kruskal-Wallis) was used to
compare lens condition discomfort ratings. Significant differences were found
between the distributions of visual discomfort ratings across conditions (p
$<$ 0.01; $\tilde{\chi}^{2}$ = 69). Bonferroni’s Post-Hoc test indicated that
the lens distortions from the lens +2D and addition +2 were more disturbing
than the rest. Likewise, the lens with a spherical power of -2 and addition +2
had the smallest discomfort. Fig 4 shows the distribution of ratings and the
statistical differences.

Figure 4: Visual discomfort ratings’ distribution across conditions, all
values have been normalised. The horizontal bars on top indicate significant
differences in Post-Hoc analysis.

The Cronbach‘s alpha value ($\rho_{\tau}$) for the rating scale was found to
be statistically consistent (0.88).

###  4.2 Gender & Age

No bias was found in the ratings of discomfort due to gender (Mann-Withney-U;
$N_{females}$ = 9 (MdN = 0.29); $N_{males}$ = 9 (MdN = 0.29); U = 51917 ;
z-val = 0.87; p = 0.39).

No correlation was found between the ratings and the age of the participants
(Spearman - $R^{2}$ = 0.0018; p = 0.97).

###  4.3 Duration

A small correlation ($R^{2}$ = 0.11, p = 0.02) was found between the duration
of the trials and the rating given. In general, subjects seemingly were faster
deciding whether to give the highest and the lowest ratings of discomfort.
However, subjects may have taken longer times during the trial if the decision
was unclear, as they perceived some level of discomfort but not the highest.

###  4.4 Yaw, pitch and roll

Tables 4, 4 and 4 present the main characteristics of the fitted distributions
for pitch, yaw and roll per trial with their mean and standard deviations. The
statistical differences of each parameter (individual, lens or rating (rounded
to the closest integer)) were computed using the Kruskal-Wallis test, and are
also reported in the tables.

Figure 5: Examples of distributions fitted for Yaw, Pitch and Roll. The green
line denotes the final distribution fitted, and the dashed red and blue lines
indicate the individual distributions (inverse power Batschelet or von Mises,
contributing to the final one). The icons on the Yaw plot represent the same
locations as in the Fig. 2)

| $-/+2$ | $-/+3$ | -2/+2 |  $+2/+2$ | Baseline | Lens |  Ratings | Subj
---|---|---|---|---|---|---|---|---
$\mu_{1}$ | $-1.65\pm 0.10$ | $-1.64\pm 0.11$ | -1.65 p m 0.12 |  $-1.65\pm 0.10$ | $-1.66\pm 0.08$ | $n.s.$ |  $n.s.$ | $***$
$\kappa_{1}$ | $\hskip 3.87498pt6.04\pm 2.46\hskip 3.87498pt$ | $\hskip 3.87498pt6.58\pm 2.88\hskip 3.87498pt$ |  6.42 p m 2.68 |  $\hskip 3.87498pt6.57\pm 2.91\hskip 3.87498pt$ | $\hskip 3.87498pt6.15\pm 2.54\hskip 3.87498pt$ | $n.s.$ |  $n.s.$ | $***$
$\lambda_{1}$ | $0.80\pm 0.22$ | $0.76\pm 0.27$ | 0.75 p m 0.26 |  $0.68\pm 0.31$ | $0.77\pm 0.23$ | $n.s.$ |  $n.s.$ | $***$
FWHM1 | $0.20\pm 0.12$ | $0.23\pm 0.17$ | 0.23 p m 0.17 |  $0.28\pm 0.20$ | $0.22\pm 0.12$ | $**$ |  $***$ | $***$
$\omega$ | $0.59\pm 0.09$ | $0.58\pm 0.11$ | 0.59 p m 0.10 |  $0.60\pm 0.10$ | $0.59\pm 0.09$ | $n.s.$ |  $n.s.$ | $***$
$\mu_{2}$ | $1.75\pm 0.13$ | $1.77\pm 0.13$ | 1.75 p m 0.14 |  $1.77\pm 0.17$ | $1.76\pm 0.16$ | $n.s.$ |  $n.s.$ | $***$
$\kappa_{2}$ | $2.10\pm 0.76$ | $2.17\pm 0.99$ | 2.26 p m 0.96 |  $2.25\pm 0.84$ | $2.10\pm 0.72$ | $n.s.$ |  $n.s.$ | $***$
$\lambda_{2}$ | $1.00$ | $1.00$ | 1.00 |  $1.00$ | $1.00$ | $n.s.$ |  $n.s.$ | $***$
FWHM2 | $0.34\pm 0.18$ | $0.35\pm 0.21$ | 0.32 p m 0.18 |  $0.32\pm 0.23$ | $0.34\pm 0.18$ | $n.s.$ |  $n.s.$ | $***$

Table 2: Yaw. Two inverse power Batschelet $\mathcal{F}$ PB were fitted. In
addition to those in the caption of Table 2, $\lambda$ refers to the peakness
of the fitted distribution.

1-9 | $-/+2$ | $-/+3$ | -2/+2 |  $+2/+2$ | Baseline | Lens |  Ratings | Subj
---|---|---|---|---|---|---|---|---
$\mu_{1}$ | $-0.05\pm 0.39$ | $-0.06\pm 0.39$ | -0.03 p m 0.48 |  $-0.10\pm 0.42$ | $-0.01\pm 0.53$ | $n.s.$ |  $*$ | $***$
$\kappa_{1}$ | $277.66\pm 145.80$ | $286.41\pm 160.89$ | 283.69 p m 148.15 |  $296.62\pm 150.39$ | $327.90\pm 155.98$ | $n.s.$ |  $*$ | $***$
FWHM1 | $0.55\pm 0.19$ | $0.55\pm 0.22$ | 0.55 p m 0.21 |  $0.57\pm 0.21$ | $0.54\pm 0.20$ | $n.s.$ |  $n.s.$ | $***$
$\omega$ | $0.19\pm 0.22$ | $0.20\pm 0.20$ | 0.15 p m 0.18 |  $0.19\pm 0.20$ | $0.17\pm 0.19$ | $n.s.$ |  $n.s.$ | $***$
$\mu_{2}$ | $-0.33\pm 0.15$ | $-0.33\pm 0.17$ | -0.31 p m 0.14 |  $-0.32\pm 0.14$ | $-0.33\pm 0.16$ | $n.s.$ |  $*$ | $***$
$\kappa_{2}$ | $48.66\pm 90.70$ | $49.47\pm 83.45$ | 45.12 p m 88.38 |  $44.28\pm 83.99$ | $48.41\pm 79.49$ | $n.s.$ |  $n.s.$ | $***$
FWHM2 | $0.52\pm 0.31$ | $0.45\pm 0.29$ | 0.57 p m 0.32 |  $0.49\pm 0.29$ | $0.50\pm 0.36$ | $n.s.$ |  $n.s.$ | $***$

Table 3: Pitch. Two Von Mises ($\mathcal{M}$) were fitted. For each of them
$\mu$ stands for mean direction, $\kappa$ is concentration, $\omega$ is the
contribution of the function from 0 to 1, and FWHM stands for full-width half
maximum of the distribution.

| $-/+2$ | $-/+3$ | -2/+2 |  $+2/+2$ | Baseline | Lens |  Ratings | Subj
---|---|---|---|---|---|---|---|---
$\mu$ | $0.02\pm 0.04$ | $0.02\pm 0.05$ | 0.03 p m 0.04 |  $0.02\pm 0.04$ | $0.03\pm 0.04$ | $n.s.$ |  $n.s.$ | $***$
$\kappa$ | $344.33\pm 142.43$ | $352.14\pm 128.91$ | 360.01 p m 130.41 |  $361.83\pm 125.27$ | $382.43\pm 127.06$ | $n.s.$ |  $**$ | $***$
FWHM | $0.14\pm 0.04$ | $0.14\pm 0.04$ | 0.13 p m 0.04 |  $0.13\pm 0.03$ | $0.13\pm 0.03$ | $n.s.$ |  $**$ | $***$

Table 4: Roll. A single Von Mises distribution ($\mathcal{M}$).

None of the parameters above mentioned was significantly different across lens
conditions, but the full-width half maximum (FWHM) of the first inverse power
Bachelet distribution on yaw. The same FWHM, as well as the mean directions of
pitch, and the FWHM of the roll changed across rating groups.

###  4.5 Gaze behaviour

##### Fixations

The number of fixations per minute and duration of fixations were
significantly different across subjects (Kruskal-Wallis, H(17) = 255, p $<$
0.001) and (H(17) = 197, p $<$ 0.001). Significant differences were also found
in the number of fixations per minute across ratings (KW, H(5) = 25.7,
p$<$0.001), where the greater discomfort is perceived, the higher amount of
fixations are found, with the exception of the group with the highest
discomfort (4-5 ratings).

##### Saccades

The number of saccades per minute, as well as the peak velocity and the
average distance, travelled on a saccade, were significantly different across
participants (Kruskal-Wallis, H(17) = 269, p $<$ 0.001),(KW, H(17) = 301, p
$<$ 0.001) and (KW, H(17) = 285, p $<$ 0.001). In the trials with more
discomfort, the saccades were on average larger (KW, H(5) = 14, p $<$ 0.05)
and had a higher peak velocity (KW, H(5) = 18, p $<$ 0.01). Significantly (KW,
H(4) = 9.6, p $<$ 0.05) more saccades were performed when the lens presented
had higher amount of distortions.

##### Observed displacement, magnification, rotation, skew and aspect

All the components that defined the experienced distortion computed from the
movement data of each participant and the lens worn were significantly
different (KW, H(4) < 434 , p $<$ 0.001) for every lens condition tested in
their mean and standard deviation. The ranking order for it can be observed in
Table 5. Significantly more mean pixel displacement, magnification,
differences on aspect ratio, skew, and rotation were observed when the
subjects indicated higher discomfort (KW, H(5) < 103 , p $<$ 0.001). The
standard deviation of all the observed parameters was significantly higher
when more discomfort was perceived. No differences across subjects were found
for any of the mean values or standard deviation, but rotation standard
deviation differences across subjects was close to significant (KW, H(17) =
27, p = 0.057).

Parameter | ${}^{+2}/_{+2}$ | ${}^{-}/_{+3}$ | ^-/_+2 |  ${}^{-2}/_{+2}$
---|---|---|---|---
Displacement Mean |  |  |  |
Displacement SD |  |  |  |
Magnification Mean |  |  |  |
Magnification SD |  |  |  |
Aspect Mean |  |  |  |
Aspect SD |  |  |  |
Skew Mean |  |  |  |
Skew SD |  |  |  |
Rotation Mean |  |  |  |
Rotation SD |  |  |  |
Table 5: Matrix draw showing the lens conditions ordered from highest (darker)
to lowest(brighter), if the same colour appeared no significant differences
were found between these two conditions.

##  5 Model - Gradient Boosting Regressor

Understanding what features might have influenced the decision for specific
discomfort ratings can be a herculean task if deferred to traditional methods.
However, thanks to the advances in computational power and machine learning
algorithms, it is possible nowadays. Using scikit learn package in Python
(Pedregosa et al., 2011), we built a gradient boosting regressor (Friedman,
2002), which ensembles several decision trees, using the residuals to fit the
next iteration and weighting them to obtain a final model, capable of not only
predicting new ratings given the estimators but also providing the relevance
of the estimators.

The dataset used comprised a total of 450 trials, and contained age, gender,
duration, baseline, maximum and minimum ratings of the subject, absolute mean
and standard deviations of the "foveally" observed factors as described
before, each of the parameters fitted to pitch, yaw and roll, and the already
analysed data from the eye-tracking (i.e. fixations and saccades).

70% of the dataset was used to train the model, and the remaining 30% was only
used to test the model’s accuracy, i.e. the data was never shown to the model
until the testing phase. The split was performed with stratification on
subjects and lens conditions and testing for its robustness. The best hyper-
parameters were decided using a bayesian optimisation search from skopt (Head
et al., 2020), and a repeated stratified group cross-validation fold within a
pipeline.

The best features were selected from the model using the meta-transformer
SelectFromModel and their relevance to the model was estimated using the
"SHAP" or SHapley Additive exPlanation values (Lundberg and Lee, 2017).

Figure 6: Visual discomfort ratings’ distribution across lenses, all values
have been normalised. Different hues show the training dataset(true values),
test dataset and the predictions from the model.

Over 400 seeds of random splits were tested to assure the model robustness,
the table 5 shows the results of it.

| $\mathcal{R}^{2}_{test}$ | $\mathcal{R}^{2}_{train}$ | $MAPE$ |  MSE | Maximum
---|---|---|---|---|---
error |  |  |  |  |
Mean | 0.53 | 0.84 | $44.18\%$ |  0.33 | 2.12
SD | $\pm 0.08$ | $\pm 0.06$ | p m 4.69% |  $\pm 0.05$ | $\pm 0.30$

Table 6: Performance metrics from the model.

Fig. 6 presents the predictions of the model classified for different lenses
and compared with the real values. Fig. 7 shows the residuals of the model for
the train and test dataset.

Figure 7: Residuals of the model depicted using Bengfort and Bilbro (2019)
yellowbrick library for one of the best outcomes of the model.

The Fig. 8 presents the "SHAP" values for the selected features of the model.

Figure 8: SHAP values for the selected features of the model.

##  6 Discussion

This study shows that distortions of PALs can be simulated in Virtual Reality
and that different lenses holding various degrees of distortions are perceived
by the participants as having different levels of vexation, being those that
presented a higher level of distortions experienced as more uncomfortable. In
our case, the lens with spherical power $+2\text{\,}\mathrm{D}$ and addition
$+2\text{\,}\mathrm{D}$ was rated the worst of all, followed by those without
spherical power and addition $+3\text{\,}\mathrm{D}$ and
$+2\text{\,}\mathrm{D}$. One would expect the baseline (i.e. no distortions)
to be rated as the most comfortable, but nevertheless, the lens distortions
due to the negative spherical power and the +2 addition presented levels of
discomfort equal to the baseline. A sum of several factors could have
contributed to this result. On the one hand, these lens distortions have low
levels of displacement in the central area, as can be seen in Fig. 3, which
could explain similar ratings to the baseline. We also cannot be entirely sure
that the manufacturer has fully corrected in the software the distortions
caused by the VR headset lenses, and considering this headset uses a canted
display, magnification from negative lenses may have made the image look more
natural. The subjective input scale was consistent given the Cronbach’s index.

Although no correlation was found between the age of the subjects and the
ratings given, it is important to mention that we have a reduced sample of
young participants with a small variance. It can not be discarded that ratings
might be different if an older group was tested, including presbyopes.
Moreover, all of our subjects were emmetropes with an aim to avoid bias in our
reduce sample, but usual ophthalmic wearers might like more the distortions
that look alike to those they are habituated to.

Subjects’ heading and gazing presented a significant variance across
individuals, independently of the distortions applied, which is expected as it
is known that these parameters are tied to each individual. Only one of the
inverse power Batschelet distributions fitted for yaw varied significantly
between the lenses and ratings, with a wider "+2/Add2" distribution, implying
the more distortions, the less accurate our horizontal head movement is. The
statistical results comparing against clusters of ratings should be taken with
care. Although non-parametric tests were used, the number of measurements on
the worse/higher ratings (4-5, after normalising) were quite low. For example,
some parameters of yaw and roll were significantly different across rating
groups, but no clear pattern was found, and no additional differences were
found in the post-hoc analysis.

Regarding gaze movements, fewer fixations per minute lead to better ratings,
this was not found between lens conditions, where there were no statistical
differences, but +2/Add2 presented more fixations per minute. In this case, a
pattern could be observed from the ratings, where the more fixations are made
per minute, the more distress is perceived, and only in the 4-5 groups it was
disrupted, probably due to the reduced sample of this groups. Thus, more
distortions may have led to more fixations, and the more one fixates, the more
one perceives those distortions, giving them worse ratings. Furthermore, more
saccades were performed in the textures with greater distortions, and ratings
followed the same pattern but did not reach significance. Otherwise, the worse
ratings presented shorter saccadic lengths and slower peak velocities in the
saccades, but, in this case, nothing could be found for the different lens
conditions. In general, gazing behaviour is highly tied to each individual,
like the heading movements. However, more saccades and more fixations led to
higher rates of discomfort, although that might have been coupled with the
lens in use.

The lens condition dictated how much of the mean and standard deviation of
displacement, magnification, rotation, differences in the aspect ratio, and
skew were observed, a factor which influence the final rating. No differences
across subjects were found, probably because even if they gaze differently,
the area they cover and locations they gaze through were pretty similar.

### Limitations of the study

There are several limiting factors that may have constrained the study’s
outcome. One of the main hiccups of the data was the reduced amount of
high/worse discomfort ratings (see Fig. 7), precluding the drawing of some
conclusions and probably limiting the model’s accuracy. To gather more data
with high ratings, the sample size must be increased by presenting more trials
with higher distortions lenses, more lens conditions with a high amount of
distortions, or simply measuring more subjects. In fact, including more
lenses, with not only different prescription powers but also different designs
may provide more information on individual characteristics and their influence
on discomfort.

In this study, only "static distortions" were measured, i.e. the distortions
of the PALs were only measured on the lens surface and presented as if looking
through the main line of sight. Although the perceived distortions varied when
looking through different areas, the distortions applied through the custom
shader did not update with the gaze. Presenting "dynamic distortions" entails
having previously measured and stored a texture of how each pixel shifts on
the texture at each possible gaze point. Considering the FoV reported for this
HMD, one would need roughly 18000 measurements per lens to do it on every
degree of the visual field. This task becomes unfeasible unless a model is
built that, from lesser measurements, interpolates the rest of the textures.

As already mentioned, the lenses used by the HMD present several distortions
that must be corrected through software. Therefore, the image presented on
display is already warped to compensate for the lenses. It sets a limitation
as only the manufacturer usually knows how well it is corrected. Additionally,
this compensation only takes one fixed pupil position (’static distortions’),
and only a recent paper (Chan et al., 2022) has started to look at how dynamic
distortions affect the whole VR experience and discomfort by inducing
unintended optic flow.

Beyond distortions, progressive lenses present different blurring across the
visual field, usually requiring a change in gaze behaviour to avoid blurry
vision, which can additionally influence the comfort while wearing these types
of lenses. In fact, the visual system is thought to be more tolerant to
distortions than it is to blur (Barbero and Portilla, 2015). Nevertheless, a
future study should assess the discomfort that these lenses present due to
blurring alone and perhaps a combination of both.

### The model

A gradient boosting regressor model was built. Although the model’s accuracy
on a single prediction is not outstanding, the model does behave as expected
for the different lenses. It is capable of predicting ratings well within
ranges defined for each lens condition. As mentioned in the limitations, the
model could be further enhanced if more data was acquired, mostly to support
the high discomfort ratings.

Although SHAP values (Lundberg and Lee, 2017) were calculated to look for how
model features act on predictions of discomfort, these values should be taken
with care, especially when trying to understand decision tree models. The
minimum rating given by the subject can be found as a main contributor. This
could have occurred because the model may have learned to identify the subject
from the minimum rating given, hence putting all individual preference weights
into that decision, i.e. basing its decisions on its idea of which subject was
estimating, even though this information was not. Other factors contributing
to the predictions were the baseline values reported by the subject, which
also could have served to acknowledge the participant, the SD of the observed
skew, aspect ratio differences, magnification and a minor role of
displacement. The concentration parameter of the first distribution fitted in
yaw also contributed, which might be an indirect consequence of how straight
was the head relative to a gazed target, upon a yaw movement, due to the
different lenses.

It is not strange that skew was found to be a contributing factor to
discomfort while "wearing" PALs. In fact, other studies already connected skew
and adaptation (Habtegiorgis et al., 2017; Rifai et al., 2020). Magnification
was also previously connected to the so-called swim effect, which relates to
the illusory and variable seesaw-like movement of the visual field that
originates with lateral head movements. This is further perceived with the
nonuniform optical magnification effects between near and far objects (Han et
al., 2003). Magnification in the adaptation to spectacles was also mentioned
to induce changes in the vestibular ocular reflex and discomfort (Cannon et
al., 1985), which might induce more discomfort in VR (Chang et al., 2020).

##  7 Conclusions

Similar to how using free-form technologies impacted the PALs market, reducing
stock, costs, and expanding the fitting possibilities (Alonso et al., 2019),
virtual reality holds the potential to become a testing bench for acceptance
of new optical consumer solutions and thus to decrease the innovation costs of
developing such lenses. VR combines the naturality of performing a daily task
with the ability to tweak specific conditions which might not be possible in
the real world.

This study proves it is possible to simulate, in virtual reality (up to a
certain degree), how distortions affect our visual field given different
progressive lenses. Besides the high inter-subject variability, every
participant perceived different levels of discomfort for different amounts of
distortions. Other traits were found beyond subjects’ susceptibility or the
distortions quantity defining the discomfort level that one can perceive with
a specific lens in certain conditions, such as certain gazing or heading
behaviours.

Finally, a machine learning model may help reduce the amount of testing
required and help to understand what features contribute the most or why this
sensation of discomfort does appear for some but not everyone. However, an
ampler amount of data is required to obtain a more applicable model.

##  8 Acknowledgements

The authors acknowledge support by the Open Access Publishing Fund of the
University of Tübingen and the state of Baden-Württemberg through bwHPC.

##  9 Declarations

## Funding

This work was supported by the European Grant PLATYPUS (Grant Agreement No
734227), a Marie Sklodowska-Curie RISE initiative. Authors MGG & YS are
employees of the University of Tübingen (E), SW is employed by Carl Zeiss
Vision International GmbH (E) and is a scientist at the University Tübingen.
TW is employed (E) by Western Sydney University. According to the journal
policy, they declare their employment positions.

The founders did not play any additional role in the study design, data
collection, and analysis, decision to publish, or preparation of the
manuscript. The specific roles of these authors are articulated in the ’author
contributions’ statement.

## Conflicts of interest/Competing interests

The authors declare that the research was conducted in the absence of any
commercial or financial relationships that could be construed as a potential
conflict of interest.

## Availability of data and material (data transparency)

The datasets generated during and/or analysed during the current study are
available from the corresponding author on reasonable request.

## Authors’ contributions

Conceptualization: MGG, TW & SW. Formal analysis, methodology, software,
validation and visualization: MGG & YS. Data curation, investigation, writing-
original draft: MGG. Writing-review and editing: MGG, YS, TW & SW.
Supervision, resources and project administration: TW & SW. Funding
acquisition: SW.

## Ethics

The ethics authorisation to perform the measurements was granted by the Ethics
Committee at the Medical Faculty of the Eberhard-Karls University and the
University Hospital Tübingen with the ID 986/2020BO2.

## References

  * Ahmad Najmee et al. [2017] Nur Aresya Ahmad Najmee, Noor Halilah Buari, Rabiatun Mujari, and Muhammad Irwan Rahman.  Satisfaction Level of Progressive Additional Lens (PALs) Wearers.  _Environment-Behaviour Proceedings Journal_ , 2(6):373, 11 2017.  doi: 10.21834/e-bpj.v2i6.999.
  * Alonso et al. [2019] José Alonso, José A. Gómez-Pedrero, and Juan A. Quiroga.  _Modern Ophthalmic Optics_.  Cambridge University Press, 3 2019.  ISBN 9781316275474.  doi: 10.1017/9781316275474.  URL https://www.cambridge.org/core/product/identifier/9781316275474/type/book.
  * Alvarez et al. [2009] Tara L. Alvarez, Sang Han, Crystal Kania, Eun Kim, Oscar Tsang, John L. Semmlow, Berangere Granger-Donetti, and Claude Pedrono.  Adaptation to progressive lenses by presbyopes.  In _2009 4th International IEEE/EMBS Conference on Neural Engineering_ , pages 143–146. IEEE, 4 2009.  ISBN 978-1-4244-2072-8.  doi: 10.1109/NER.2009.5109255.
  * Alvarez et al. [2017] Tara L Alvarez, Eun H Kim, and Bérangère Granger-Donetti.  Adaptation to Progressive Additive Lenses: Potential Factors to Consider.  _Scientific Reports_ , 7(1):2529, 2017.  ISSN 20452322.  doi: 10.1038/s41598-017-02851-5.  URL http://www.ncbi.nlm.nih.gov/pubmed/28566706http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=PMC5451391.
  * Atchison [1987] David A. Atchison.  Optical performance of progressive power lenses.  _Clinical and Experimental Optometry_ , 70(5):149–155, 9 1987.  ISSN 0816-4622.  doi: 10.1111/j.1444-0938.1987.tb04235.x.
  * Aves [1907] Owen Aves.  Special properties achieved by the combination of the front and back surfaces, 1907.  URL https://patents.google.com/patent/GB190715735A/en?q=spectacle+lens&inventor=owen+aves&after=priority:19070101&scholar.
  * Barbero and González [2020] Sergio Barbero and María del Mar González.  Admissible surfaces in progressive addition lenses.  _Optics Letters_ , 45(20), 10 2020.  ISSN 0146-9592.  doi: 10.1364/OL.401927.
  * Barbero and Portilla [2015] Sergio Barbero and Javier Portilla.  Geometrical interpretation of dioptric blurring and magnification in ophthalmic lenses.  _Optics Express_ , 23(10):13185, 5 2015.  ISSN 1094-4087.  doi: 10.1364/oe.23.013185.
  * Bengfort and Bilbro [2019] Benjamin Bengfort and Rebecca Bilbro.  Yellowbrick: Visualizing the Scikit-Learn Model Selection Process.  _Journal of Open Source Software_ , 4(35):1075, 3 2019.  ISSN 2475-9066.  doi: 10.21105/joss.01075.
  * Bist et al. [2021] Jeewanand Bist, Dinesh Kaphle, Sanjay Marasini, and Himal Kandel.  Spectacle non-tolerance in clinical practice – a systematic review with meta-analysis.  _Ophthalmic and Physiological Optics_ , 41(3):610–622, 5 2021.  ISSN 0275-5408.  doi: 10.1111/opo.12796.
  * Cannon et al. [1985] Stephen C. Cannon, R. John Leigh, David S. Zee, and Larry A. Abel.  The Effect of the Rotational Magnification of Corrective Spectacles on the Quantitative Evaluation of the VOR.  _Acta Oto-Laryngologica_ , 100(1-2):81–88, 1 1985\.  ISSN 0001-6489.  doi: 10.3109/00016488509108591.
  * Chan et al. [2022] Tsz Tai Chan, Yixuan Wang, Richard Hau Yue So, and Jerry Jia.  Predicting Subjective Discomfort Associated with Lens Distortion in VR Headsets During Vestibulo-Ocular Response to VR Scenes.  _IEEE Transactions on Visualization and Computer Graphics_ , pages 1–1, 2022.  ISSN 1077-2626.  doi: 10.1109/TVCG.2022.3168190.
  * Chang et al. [2020] Eunhee Chang, Hyun Taek Kim, and Byounghyun Yoo.  Virtual Reality Sickness: A Review of Causes and Measurements.  _International Journal of Human-Computer Interaction_ , 36(17):1658–1682, 2020.  ISSN 15327590.  doi: 10.1080/10447318.2020.1778351.  URL https://doi.org/10.1080/10447318.2020.1778351.
  * Cho et al. [1991] M H Cho, C H Spear, and L Caplan.  The effect of excessive add power on the acceptance of progressive addition lenses.  _Journal of the American Optometric Association_ , 62(9):672–5, 9 1991.  ISSN 0003-0244.
  * Dai et al. [2016] Weiwei Dai, Ivan Selesnick, John-Ross Rizzo, Janet Rucker, and Todd Hudson.  A parametric model for saccadic eye movement.  In _2016 IEEE Signal Processing in Medicine and Biology Symposium (SPMB)_ , pages 1–6. IEEE, 12 2016.  ISBN 978-1-5090-6713-8.  doi: 10.1109/SPMB.2016.7846860.
  * Esser et al. [2017] Gregor Esser, Wolfgang Becken, Helmut Altheimer, and Werner Müller.  Generalization of the Minkwitz theorem to nonumbilical lines of symmetrical surfaces.  _Journal of the Optical Society of America. A, Optics, image science, and vision_ , 34(3):441–448, 3 2017.  ISSN 1520-8532.  doi: 10.1364/JOSAA.34.000441.  URL http://www.ncbi.nlm.nih.gov/pubmed/28248371.
  * Friedman [2002] Jerome H. Friedman.  Stochastic gradient boosting.  _Computational Statistics & Data Analysis_, 38(4):367–378, 2 2002.  ISSN 01679473.  doi: 10.1016/S0167-9473(01)00065-2.
  * Habtegiorgis et al. [2017] Selam Wondimu Habtegiorgis, Katharina Rifai, Markus Lappe, and Siegfried Wahl.  Adaptation to Skew Distortions of Natural Scenes and Retinal Specificity of Its Aftereffects.  _Frontiers in Psychology_ , 8:1158, 7 2017.  ISSN 1664-1078.  doi: 10.3389/fpsyg.2017.01158.  URL http://journal.frontiersin.org/article/10.3389/fpsyg.2017.01158/full.
  * Han et al. [2003] Ying Han, Kenneth J. Ciuffreda, Arkady Selenow, and Steven R. Ali.  Dynamic interactions of eye and head movements when reading with single-vision and progressive lenses in a simulated computer-based environment.  _Investigative Ophthalmology and Visual Science_ , 44(4):1534–1545, 4 2003.  ISSN 01460404.  doi: 10.1167/iovs.02-0507.  URL http://iovs.arvojournals.org/article.aspx?doi=10.1167/iovs.02-0507.
  * Head et al. [2020] Tim Head, Manoj Kumar, Holger Nahrstaedt, Gilles Louppe, and Iaroslav Shcherbatyi.  scikit-optimize/scikit-optimize, 9 2020.  URL https://doi.org/10.5281/zenodo.4014775.
  * Hutchings et al. [2007] Natalie Hutchings, Elizabeth L. Irving, Nadine Jung, Lisa M. Dowling, and Kelly A. Wells.  Eye and head movement alterations in naïve progressive addition lens wearers.  _Ophthalmic and Physiological Optics_ , 27(2):142–153, 3 2007.  ISSN 02755408.  doi: 10.1111/j.1475-1313.2006.00460.x.
  * Lundberg and Lee [2017] Scott M Lundberg and Su-In Lee.  A Unified Approach to Interpreting Model Predictions.  In I Guyon, U Von Luxburg, S Bengio, H Wallach, R Fergus, S Vishwanathan, and R Garnett, editors, _Advances in Neural Information Processing Systems_ , volume 30. Curran Associates, Inc., 2017.  URL https://proceedings.neurips.cc/paper/2017/file/8a20a8621978632d76c43dfd28b67767-Paper.pdf.
  * Maitenaz [1969] Bernard F. Maitenaz.  Ophthalmic lenses with a progressively varying focal power, 1969.
  * Minkwitz [1963] G. Minkwitz.  Über den Flächenastigmatismus Bei Gewissen Symmetrischen Asphären.  _Optica Acta: International Journal of Optics_ , 10(3):223–227, 7 1963.  ISSN 0030-3909.  doi: 10.1080/713817794.  URL https://www.tandfonline.com/doi/full/10.1080/713817794.
  * Mulder et al. [2020] Kees Mulder, Irene Klugkist, Daan van Renswoude, and Ingmar Visser.  Mixtures of peaked power Batschelet distributions for circular data with application to saccade directions.  _Journal of Mathematical Psychology_ , 95:102309, 4 2020\.  ISSN 00222496.  doi: 10.1016/j.jmp.2019.102309.
  * Mulder [2019] Kees Tim Mulder.  _Bayesian Circular Statistics: von Mises-based solutions for practical problems_.  PhD thesis, Utrecht University, 6 2019.
  * Ogle and Saunders W.B. [1950] Kenneth Neil Ogle and Saunders W.B.  _Researches in binocular vision_.  W.B .Saunders, Philadelphia, 1950.  URL https://books.google.de/books?id=PykEAQAAIAAJ.
  * Pedregosa et al. [2011] Fabian Pedregosa, Gaël Varoquaux, Alexandre Gramfort, Vincent Michel, Bertrand Thirion, Olivier Grisel, Mathieu Blondel, Peter Prettenhofer, Ron Weiss, Vincent Dubourg, Jake Vanderplas, Alexandre Passos, David Cournapeau, Matthieu Brucher, Matthieu Perrot, and Edouard Duchesnay.  Scikit-learn: Machine Learning in Python.  _Journal of Machine Learning Research_ , 12(85):2825–2830, 2011.  URL http://jmlr.org/papers/v12/pedregosa11a.html.
  * Poullain and Cornet [1911] Augustin Georges Poullain and Darius Henri Julien Cornet.  Optical lens., 7 1911.  URL https://patents.google.com/patent/US1143316A/en?oq=U.S.+Patent+No.+1%2C143%2C316.
  * Rifai et al. [2020] Katharina Rifai, Selam W. Habtegiorgis, Caroline Erlenwein, and Siegfried Wahl.  Motion-form interaction: Motion and form aftereffects induced by distorted static natural scenes.  _Journal of Vision_ , 20(13):10, 12 2020.  ISSN 1534-7362.  doi: 10.1167/jov.20.13.10.
  * Sauer et al. [2020] Yannick Sauer, Siegfried Wahl, and Katharina Rifai.  Parallel Adaptation to Spatially Distinct Distortions.  _Frontiers in Psychology_ , 11, 11 2020.  ISSN 1664-1078.  doi: 10.3389/fpsyg.2020.544867.
  * Sauer et al. [2022] Yannick Sauer, Alexandra Sipatchin, Siegfried Wahl, and Miguel García García.  Assessment of consumer VR-headsets’ objective and subjective field of view (FoV) and its feasibility for visual field testing.  _Virtual Reality_ , 1 2022.  ISSN 1359-4338.  doi: 10.1007/s10055-021-00619-x.  URL https://link.springer.com/10.1007/s10055-021-00619-x.
  * Sawides et al. [2010] L. Sawides, S. Marcos, S. Ravikumar, L. Thibos, A. Bradley, and M. Webster.  Adaptation to astigmatic blur.  _Journal of Vision_ , 10(12):22–22, 10 2010.  ISSN 1534-7362.  doi: 10.1167/10.12.22.  URL http://jov.arvojournals.org/Article.aspx?doi=10.1167/10.12.22.
  * Sullivan and Fowler [1988] Colin M. Sullivan and Colin W. Fowler.  Progressive addition and veriables focus lenses: A review.  _Ophthalmic and Physiological Optics_ , 8(4):402–414, 10 1988.  ISSN 0275-5408.  doi: 10.1111/j.1475-1313.1988.tb01177.x.  URL https://onlinelibrary.wiley.com/doi/10.1111/j.1475-1313.1988.tb01177.x.
  * The European Parliament and the Council of the European Union [2016] The European Parliament and the Council of the European Union.  Regulation (EU) 2016/679 of the European Parliament and of the Council.  _Official Journal of the European Union_ , 59(L119/1), 2016\.
  * van der Lans et al. [2011] Ralf van der Lans, Michel Wedel, and Rik Pieters.  Defining eye-fixation sequences across individuals and tasks: the Binocular-Individual Threshold (BIT) algorithm.  _Behavior Research Methods_ , 43(1):239–257, 3 2011.  ISSN 1554-3528.  doi: 10.3758/s13428-010-0031-2.
  * Vinas et al. [2012] Maria Vinas, Lucie Sawides, Pablo de Gracia, and Susana Marcos.  Perceptual Adaptation to the Correction of Natural Astigmatism.  _PLoS ONE_ , 7(9):e46361, 9 2012.  ISSN 1932-6203.  doi: 10.1371/journal.pone.0046361.  URL http://dx.plos.org/10.1371/journal.pone.0046361.
  * Volk and Weinberg [1962] D. Volk and J. W. Weinberg.  The Omnifocal Lens for Presbyopia.  _Archives of Ophthalmology_ , 68(6):776–784, 12 1962.  ISSN 0003-9950.  doi: 10.1001/archopht.1962.00960030780012.