Machine Learning based Predictions of Subjective

Refractive Errors of the Human Eye

Alexander Leube

1,2,*

, Christian Leibig

1,*

, Arne Ohlendorf

1,2

and Siegfried Wahl

1,2

Institute for Ophthalmic Research, Eberhard Karls University, Tübingen, Germany

Carl Zeiss Vision International GmbH, Technology and Innovation, Aalen Germany

Keywords: Big Data, Machine Learning, Subjective Refraction.

Abstract: The aim of this research was to demonstrate the suitability of a data-driven approach to identify the sphero-

cylindrical subjective refraction. An artificial deep learning network with two hidden layers was trained to

predict power vector refraction (M, J

and J

) from 37 dimensional feature vectors (36 Zernike coefficients

+ pupil diameter) from a large database of 50,000 eyes. A smaller database of 460 eyes containing subjective

and objective refraction from controlled experiment conditions was used to test for prediction power. Bland-

Altmann analysis was performed, calculating the mean difference (eg ΔM) and the 95% confidence interval

(CI) between predictions and subjective refraction. Using the machine learning approach, the accuracy

(ΔM = +0.08D) and precision (CI for ΔM = ± 0.78D) for the prediction of refractive error corrections was

comparable to a conventional metric (ΔM = +0.11D ± 0.89D) as well as the inter-examiner agreement between

optometrists (ΔM = -0.05D ± 0.63D). To conclude, the proposed deep learning network for the prediction of

refractive error corrections showed its suitability to reliably predict subjective power vectors of refraction

from objective wavefront data.

1 INTRODUCTION

The current gold standard for the measurement of

refractive errors of the eye is the subjective refraction

that aims to correct the lower aberrations of the eye in

order to provide the best retinal image quality (Goss

and Grosvenor, 1996). The intra-examiner agreement

which is defined as the agreement between different

measurements of the refractive error by the same

examiner, as well as the agreement between the

subjective correction from multiple examiners (inter-

examiner agreement) are in the range between

±0.25D (80%) and ±0.50D (95%) (Goss and

Grosvenor, 1996). The subjective ability to judge the

level of focus depends on the depth of focus of the

eye, which is known to be around ±0.30D (Leube

et al. 2016) and higher order aberrations whose

influence is modulated by the size of the pupil (Wang

et al. 2003). Other factors that can affect the

successful computation of subjective corrections is

the ability of the visual system to easily adapt to blur

and contrast. For example, it is well known that the

eye is adapted to its own aberrations and can easily

Both authors contributed equally to this work.

adapt to spherical as well as astigmatic defocus (Artal

et al., 2004; Ohlendorf and Schaeffel, 2009;

Ohlendorf et al., 2011).

Pioneered by Applegate, Williams, Thibos and

others, computational attempts have been made in

order to predict the refractive correction of sphere,

cylinder and its axis from the monochromatic lower

and higher order aberrations of the eye, using image

quality metrics (Applegate et al. 2003; Guirao and

Williams 2003; Thibos et al. 2002; Thibos et al.

2004a). In general, one can distinguish between

pupil-plane metrics that are used to optimize the

wavefront aberrations of the eye and image-plane

metrics, which are used to maximize the quality of the

retinal image in relation to the subjective best image

quality (Guirao and Williams, 2003). While

comparing pupil-plane as well as image-plane metrics

to the subjectively best correction of refractive errors

in a cohort of 147 eyes, Guirao and Williams (2003)

concluded that image-plane metrics are the better

choice to predict the subjective refraction. In support

to this findings, Thibos et al. (2004) extended the use

of pupil-plane metrics as well as image-plane metrics

Leube, A., Leibig, C., Ohlendorf, A. and Wahl, S.

Machine Learning based Predictions of Subjective Refractive Errors of the Human Eye.

DOI: 10.5220/0007254401990205

In Proceedings of the 12th International Joint Conference on Biomedical Engineering Systems and Technologies (BIOSTEC 2019), pages 199-205

ISBN: 978-989-758-353-7

199

and presented 33 different metrics. Their accuracy

(defined by the mean error of the prediction) as well

as precision (equivalent to the 95% limits of

agreement) to predict the subjective refraction in a

population of 200 eyes was shown previously (Thibos

et al. 2004). The range of the mean difference for the

estimated spherical defocus was reported to be

between -0.50D and +0.25D for all metrics, while the

limits of agreement (LoA) ranged from ±0.50D to

±1.00D.

The mapping from detailed, high-dimensional

measurements of low- to high-order aberration errors

to the best possible subjective refraction correction

includes both, optical and neural factors that are

difficult to assess with detailed physical models.

Using a training set of paired objective and subjective

measurements, the task of predicting the optimal

subjective correction from an objective measurement

is essentially a (non-linear) regression problem.

While using for example a deep learning network in

order to allow enough space for a flexible function,

the prediction of refractive corrections should be

learnable from data. Further, it should perform at least

as good as detailed physical models, while the former

does not require the detailed mechanistic knowledge

of the latter. Additionally and by construction, such a

regressor inherently captures optical and neural

factors, while it is assumed that these are consistent

across subjects. The aim of the current research was

to explore the applicability of machine learning

approaches for the prediction of the sphero-

cylindrical correction of refractive errors and

compare their performance (in terms of accuracy and

precision) against currently used objective metrics

and subjective measurements.

2 METHODS

2.1 Datasets

Three datasets were used: (1) An excerpt from a data

base of spectacle lens orders (provided by Carl Zeiss

Vision GmbH, Aalen, Germany) that included

monocular data from 50,496 eyes, measured by

various professional optometrists. Each eye in this

database was characterized by its aberrometry data

and pupil size measured with a wavefront

aberrometer (i.Profiler 1, Carl Zeiss Vision GmbH,

Aalen, Germany) and a subjective measurement of its

refractive errors. Aberrometry data were assessed up

to the 7

radial order and included refractive data

such as the spherical equivalent M from -16.00D to

+9.00D with a mean of -0.85 ± 2.67D. This large data

set was divided into a training set (32,316 samples,

coined ZV train), a developing set (8,080 samples,

coined ZV develop) and a testing set (10,100 samples,

coined ZV test). (2) A second set of monocular data

was collected independently for research purposes

with a Subjective Refraction Unit (SRU) and the

wavefront aberrometer (i.Profiler plus, Carl Zeiss

Vision GmbH, Aalen, Germnay) from 460 eyes

(Ohlendorf, Leube and Wahl 2016). The SRU

included a digital phoropter (ZEISS Visuphor 500,

Carl Zeiss Vision GmbH, Aalen, Germany) and a

LCD-screen to display optotypes (ZEISS Visuscreen

500, Carl Zeiss Vision GmbH, Aalen, Germany) with

a minimum luminance of L = 250cd/m

. SLOAN

letters were used as optotypes and were shown in an

EDTRS layout (Sloan, 1959; National Eye Institute,

1991). This dataset was used exclusively for testing

purposes (simply called EagleEye) to discriminate

against ZV test. (3) A third dataset was used to

examine the inter-examiner agreement of monocular

subjective refractions in a group of 54 eyes, measured

by two of the authors both using the standardized

procedure.

2.2 Machine Learning Methodology

An artificial deep learning network was used to

compute the best correction of the power vectors of

the refractive errors for a given set of low- and high-

order aberrations of an eye. Therefore, the target

values t drawn out of the power vector entries {M, J

} were predicted by using the set of Zernike

coefficients together with the pupil diameter as input

features for any given eye. Every eye was hence

objectively characterized by concatenating these

features into a vector 𝑥 ∈ ℝ

. The non-linear

function 𝑦 = 𝑓(𝑥, 𝜃) to predict the subjectively

assigned value 𝑡 was learned by using the respective

target values from the training data set (ZV train,

32,316 samples): The networks' trainable parameters

θ were learned by backpropagating the errors between

predictions and target values. The separate

development set of data (ZV develop, 8,080 samples)

was used to determine all necessary hyperparameters

such as the network architecture described further

below. The two separate and independent test data

sets where then used to report final performance for

the sub data set ZV test that constituted a non-

overlapping split from the same database used for

training that was comprised of 10,100 samples.

HEALTHINF 2019 - 12th International Conference on Health Informatics

200

2.3 Model Development

In principle, any (non)-linear regression model that

fits the training data and generalizes to unseen test

data is conceivable. Here, we chose deep learning

networks because their effective capacity can be

tuned flexibly to the problem at hand. Initial

experiments (data not shown) ruled out a simple

linear model and Bayesian hyperparameter

optimization using hyperopt, suggested a similar

model to the one developed here by using current best

principles (Bergstra et al. 2013; Goodfellow, et al.

2016; Shahriari et al., 2016). More specifically,

development converged to using multi-layered

feedforward networks with a single linear output unit

to jointly learn the basis functions and predictions by

minimizing the mean squared error 1/N Σ

(y-t)

between predicted and subjectively assigned sphero-

cylindrical corrections. We trained three networks in

total, one each for M, J

and J

respectively. Every

network had two hidden layers, with 64 hidden units

each and parametric rectifier nonlinearities (He et al.,

2015). Initial weights were drawn from a zero-mean

Gaussian with a standard deviation of √2/n with n the

number of incoming connections (He et al., 2015). All

data were preprocessed by subtracting the mean of

each feature dimension calculated across the training

set. The network was trained using the Adam

optimizer with a learning rate of 0.001 which was

automatically reduced by a factor of 0.2 whenever the

loss on the validation set did not improve for 10

epochs (passes through the training data) (Kingma

and Ba, 2015). To avoid overfitting, small network

weights were encouraged via global L2 regularization

(λ = 0.02) and randomly dropping 0.2 of the hidden

units (Bishop, 2006; Hinton et al., 2012; Srivastava et

al., 2014). Training was performed with a batch size

of 128 and stopped, when the loss evaluated on the

validation set did not improve for 50 epochs ("early

stopping").

2.4 Implementation

The machine learning models were developed in

Python, using the scientific computing stack and in

particular the deep learning library keras together

with tensorflow as backend (Francois, 2016; Girija,

2016). Network related computations were performed

on the graphics processing unit (GeForce GTX 1080

GPU, NVIDIA, Santa Clara, USA).

Code and models will be publicly available upon

publication under https://github.com/chleibig

/airefraction.

2.5 Validation and Comparison with

Alternative Methods

To compare the performance of the machine learning

approach to (1) existing subjective measurements as

well as (2) alternative computational approaches and

(3) to the inter-examiner agreement of subjective

refractions, the following analysis was performed:

accuracy and precision in terms of Bland-Altman

analysis (Bland and Altman, 1986) of the machine

learning approach was compared to power vectors of

refraction (Thibos et al. 1997) from (1) subjective

refractions and (2) computations based corrections

using the visual Strehl of the optical transfer function

(VSOTF) (Thibos et al., 2004a), both for the

described dataset EagleEye. Inter-examiner

agreement was established for a smaller set of eyes

(dataset 3, n = 54 eyes) in order to be able to better

rank the obtained results in the framework of the

subjective assessment of refractive errors.

3 RESULTS

3.1 Testing of the Deep Learning

Network

Using the sub dataset ZV train, the deep neuronal

network was trained and further tested, while using

the additional sub datasets ZV develop. In a first step,

the performance in terms of accuracy and precision of

the deep learning network was assessed, while using

the data-set ZV test. The results revealed mean

differences (accuracy) that were around zero for the

three power vectors (M, J

and J

), while the 95%

limits of agreement (precision) were twice as big for

M as for the cylindrical vector components J

and J

(see Table 1). In a second step, the true generalization

of the deep learning network was analyzed, while

using the independent dataset EagleEye and again,

mean differences were around zero for the three

power vectors M, J

and J

, while the 95% limits of

agreement was higher for M, when compared to J

and J

. Since the developed deep learning network

performed similar on both datasets (ZV test and

EagleEye), it will be also applicable to data from

potentially different distributions.

Machine Learning based Predictions of Subjective Refractive Errors of the Human Eye

201

Table 1: Mean differences and 95% limits of agreement between computationally and subjectively assessed refraction for the

power vector components (M, J0 and J45).

Refractive

component

Mean difference,

D (95% CI)

95% limits of

agreement, lower, D

(95% CI)

95% limits of

agreement, upper, D

(95% CI)

Deep network

(ZV test, n = 10100)

0.01 (0.02, 0.01)

-0.64 (-0.66, -0.63)

0.67 (0.66, 0.68)

0.01 (0.01, 0.00)

-0.31 (-0.32, -0.30)

0.32 (0.31, 0.33)

0.00 (0.00, 0.00)

-0.28 (-0.29, -0.27)

0.28 (0.27, 0.29)

Deep network

(EagleEye, n = 460)

0.08 (0.12, 0.05)

-0.70 (-0.76, -0.63)

0.86 (0.80, 0.92)

-0.01 (0.00, 0.00)

-0.31 (-0.37, -0.25)

0.28 (0.22, 0.35)

0.01 (0.02, 0.00)

-0.23 (-0.29, -0.17)

0.25 (0.19, 0.31)

Visual Strehl

OTF (EagleEye,

n = 460)

0.11 (0.15, 0.06)

-0.78 (-0.86, -0.71)

1.00 (0.92, 1.07)

-0.02 (0.00, -0.04)

-0.40 (-0.47, -0.33)

0.36 (0.29, 0.43)

0.01 (-0.01, 0.02)

-0.27 (-0.34, -0.20)

0.28 (0.21, 0.35)

Human1 vs.

Human2 (EagleEye,

n = 53)

-0.05 (-0.13, 0.04)

-0.68 (-0.84, -0.53)

0.59 (0.43, 0.74)

0.01 (-0.02, 0.04)

-0.19 (-0.34, -0.04)

0.21 (0.06, 0.36)

-0.01 (-0.04, 0.01)

-0.17 (-0.32, -0.02)

0.14 0.01, 0.29)

3.2 Computation of Refractive Vector

Components

To compare the proposed deep learning network

approach to other computational methods, the three

vector components of refraction of dataset EagleEye

were computed using the deep learning network and

the visual metric VSOTF and judged in respect of

agreement and precision with the available subjective

data (Thibos et al. 2004). The deep learning network

performed similar in terms of agreement and

precision for all three vector components in

comparison to the traditional metric (see Table 1). In

terms of inter-examiner agreement between

professional optometrists (see Table 1) that was

calculated from double ratings of the same

participants from two optometrists in 54 eyes from

dataset 3, the artificial neural network covers a similar

range of agreement and precision for all three power

vectors of refraction.

4 DISCUSSION

In the current study, an artificial deep learning

network approach was used in order to learn the

complicated transfer from objective to subjective data

and highly non-linear mapping from objective

aberrometry data to the subjectively optimal sphero-

cylindrical correction of refractive errors was applied.

Given the results, the detailed forward modeling of

optical, neuronal and perceptional contributions to the

transfer function using a deep learning network is

comparable to earlier introduced methods (such as

visual Strehl metrics) and the current gold standard,

the subjective refraction.

4.1 Comparison with Conventional

Computational Metrics

Compared to various pupil-plane and image quality

metrics that were developed and applied for equal

objectives, our results showed similar mean

differences and comparable 95% limits of agreement,

when compared to the VSOTF metric (Thibos et al.

2004). For the prediction of M from 200 eyes,

different used metrics resulted in a mean difference

between 0.24D to -0.04D, but were not the best

ranked regarding their 95% limits of agreement. Best

results in the precision of the prediction of M was

shown from two pupil fraction metrics: PFWc and

PFSc; two image quality metrics for grating objects:

AreaOTF, the visual Strehl of the optical transfer

function (VSOTF) and one contrast metric: light-in-

the-bucket (LIB) that all ranged between ± 0.49D to

± 0.59D (Thibos et al. 2004). Using pupil plane and

image plane metrics, Guirao et al. (2003) reported a

mean difference for M of 0.40D, when predicted by

HEALTHINF 2019 - 12th International Conference on Health Informatics

202

the metrics and compared to subjective data of 146

eyes (95% limit of agreement: ± 0.98D). With respect

to the 95% limits of agreement, the results of the

current investigation revealed similar or better values,

when compared to the best ranked metrics regarding

the prediction of M by Thibos et al. (2004) as well as

Guirao et al. (2003). The average mean differences

for the spherical equivalent refractive error M is much

lower compared to the typical steps in subjective

refraction, where errors are corrected in steps of

0.25D. Therefore, the performance for the prediction

of the power vector M, J

, J

of the developed

artificial deep learning network can be classified as

excellent, with the advantage over existing

computational methods that computational time is

significantly reduced.

4.2 Agreement with Subjective

Refraction

The 95% limits of agreement for the computation of

M were similar compared to studies that subjectively

assessed the refractive to test the inter- as well as

intra-observer agreement. Zadnik et al. (1992)

reported 95% limit of agreement range from ± 0.94D

for cycloplegic subjective refraction to ± 0.63D for

the non-cycloplegic assessment of the spherical

equivalent refractive error, when refraction was

assessed by the same examiner multiple times

(Zadnik et al. 1992). In case of retinoscopy, the 95%

limits of agreement were reported to be ± 0.95D for

cycloplegic retinoscopy and ± 0.78D for non-

cycloplegic retinoscopy (Zadnik et al. 1992).

Bullimore et al. (1993) found a 95% limit of

agreement regarding the repeatability of the spherical

equivalent refractive error, measured by two

optometrists of ± 0.78D with a mean difference of -

0.12D (Elliott and Bullimore, 1993). Values reported

by Rosenfield and Chiu (1995) for the 95% limit of

agreement of the inter-observer variability are

± 0.29D, when the spherical equivalent refractive

error was subjectively assessed by the same examiner

for five times. When findings for the spherical

equivalent error from the current study are compared

to the previously reported subjective measures, it can

be concluded that the presented method is as precise

and accurate as the current gold standard, the

subjective refraction.

4.3 Retinal and Neural Factors

Affecting Perception

The question arises, whether a subjective refraction,

either under monocular or binocular conditions,

would lead to the optical best correction of the

aberrations, or if it represents a correction of

aberrations that is most accepted by the wearer. In

order to minimize any possible bias, we have only

analyzed monocular measurements of refractive

errors and additionally, the subjective refraction

followed the rule "best visual acuity with maximum

plus power". Compared to previous studies that were

conducted in order to predict the best sphero-

cylindrical correction of refractive errors using either

pupil plane or image plane metrics, our approach, an

artificial deep learning network that is able to

incorporate perceptual and neural processes,

assuming that they are similar over a cohort. Given

enough capacity from the data, deep learning

networks are able to learn any function, while

alternative methods provide design transfer functions

with possibly limited capacity. These approaches use

hypotheses and limited domain knowledge about

optics and the visual system in order to determine the

best correction of an individual's aberration, with- or

without the incorporation of for instance the contrast

sensitivity of the eye. Since the deep learning network

that was applied, is able to incorporate such

processes, the network learned the mapping from

Zernike coefficients to subjectively optimal refractive

error corrections.

4.4 Monochromatic Aberrometry vs.

Polychromatic Subjective

Refraction

The prediction of refractive errors based on

aberrometry data has some shortcomings that have to

be taken into account. When measuring the refractive

errors subjectively, a polychromatic test chart (white

produced out of the red, green and blue LEDs of the

monitor) is used, while wavefront aberrations are

measured under monochromatic conditions.

Following, one has to decide, which wavelength to

use to compute the objective autrorefraction as well

as power vectors from aberrometry measurements

and the results are only valid for the specific single

wavelength. In the current analysis, the data were

analyzed for a wavelength of λ = 550nm, based on the

maximal spectral sensitivity of the eye and this is in

contradiction to for example Thibos et al., where they

have used a reference wavelength of λ = 570nm

(Thibos et al. 2004). Since the obtained results are

comparable to the data of the subjective refractions,

we conclude that the chosen wavelength only

minimally effects the results obtained.

Machine Learning based Predictions of Subjective Refractive Errors of the Human Eye

203

5 CONCLUSIONS

The aim of the current research was to use artificial

deep learning networks for the prediction of

subjective refractive corrections, in order to allow for

an effective description of perceptual and neural

processes that occur during the subjective assessment

of such errors. The obtained results have shown that

the presented methods lead to exact values of the

power vectors of refraction, when compared to the

subjective measurement and to a conventional metric.

Additionally, aberrations need not necessarily be

described by Zernike coefficients, neither is a detailed

description more powerful to predict the refractive

errors.

ACKNOWLEDGEMENTS

The authors would like to thank Steve Spratt from the

Carl Zeiss Vision International GmbH to provide and

support with the objective and subjective refraction

database.

REFERENCES

Applegate, R. A. et al. (2003) Interaction between

aberrations to improve or reduce visual performance,

Journal of Cataract and Refractive Surgery, 29(8),

1487–1495.

Applegate, R. A. et al. (2003) Visual acuity as a function of

Zernike mode and level of root mean square error,

Optometry and vision science , 80(2), 97–105.

Artal, P. et al. (2004) Neural compensation for the eye’s

optical aberrations, Journal of Vision, 4(4), 281–287.

Bergstra, J., Yamins, D. L. K. and Cox, D. D. (2013)

Making a Science of Model Search: Hyperparameter

Optimization in Hundreds of Dimensions for Vision

Architectures, ICML, 115–123. doi: 1209.5111v1.

Bishop, C. M. C. C. M. (2006) Pattern Recognition and

Machine Learning, Pattern Recognition. doi:

10.1117/1.2819119.

Bland, J. M. and Altman, D. G. (1986) Statistical methods

for assessing agreement between two methods of

clinical measurement, Lancet, 1(8476), 307–310.

Elliott, D. B. and Bullimore, M. A. (1993) Assessing the

reliability, discriminative ability, and validity of

disability glare tests, Investigative Ophthalmology &

Visual Science, 34(1), 108–119.

Francois, C. (2016) Keras, GitHub repository. Available at:

https://github.com/fchollet/keras.

Girija, S. S. (2016) Tensorflow: Large-scale machine

learning on heterogeneous distributed systems.

Goodfellow, Ian, Bengio, Yoshua, Courville, A. and

Goodfellow, I. (2016) Deep Learning, MIT Press.

Goss, D. A. and Grosvenor, T. (1996) Reliability of

refraction--a literature review, Journal of the American

Optometric Association, 67(10), 619–630.

Guirao, A. and Williams, D. R. (2003) A method to predict

refractive errors from wave aberration data, Optometry

and Vision Science, 80(1), 36–42.

He, K. et al. (2015) Delving deep into rectifiers: Surpassing

human-level performance on imagenet classification, in

Proceedings of the IEEE International Conference on

Computer Vision, 1026–1034. doi:

10.1109/ICCV.2015.123.

Hinton, G. E. et al. (2012) Improving neural networks by

preventing co-adaptation of feature detectors, arXiv

preprint arXiv:1207.0580.

Kingma, D. P. and Ba, J. L. (2015) Adam: a Method for

Stochastic Optimization, International Conference on

Learning Representations 2015, 1–15. doi:

h10.1145/1830483.1830503.

Leube, A., Ohlendorf, A. and Wahl, S. (2016) The influence

of induced astigmatism on the depth of focus,

Optometry and Vision Science, 93(10). doi:

10.1097/OPX.0000000000000961.

National Eye Institute (1991) Early Treatment Diabetic

Retinopathy Study design and baseline patient

characteristics. ETDRS report number 7,

Ophthalmology.

Ohlendorf, A., Leube, A. and Wahl, S. (2016) Steps

towards Smarter Solutions in Optometry and

Ophthalmology-Inter-Device Agreement of Subjective

Methods to Assess the Refractive Errors of the Eye,

Healthcare (Basel), 4(3). doi:

10.3390/healthcare4030041.

Ohlendorf, A. and Schaeffel, F. (2009) Contrast adaptation

induced by defocus - a possible error signal for

emmetropization?, Vision Research, 49(2), 249–256.

doi: 10.1016/j.visres.2008.10.016.

Ohlendorf, A., Tabernero, J. and Schaeffel, F. (2011)

Neuronal adaptation to simulated and optically-induced

astigmatic defocus, Vision Research, 51(6), 529–534.

doi: 10.1016/j.visres.2011.01.010.

Rosenfield, M. and Chiu, N. N. (1995) Repeatability of

subjective and objective refraction’, Optometry and

Vision Science, 72(8), 577–579.

Shahriari, B. et al. (2016) Taking the human out of the loop:

A review of Bayesian optimization, Proceedings of the

IEEE, 148–175. doi: 10.1109/JPROC.2015.2494218.

Sloan, L. L. (1959) New test charts for the measurement of

visual acuity at far and near distances, American

Journal of Ophthalmology, 48(6), 807–813.

Srivastava, N. et al. (2014) Dropout: A Simple Way to

Prevent Neural Networks from Overfitting, Journal of

Machine Learning Research, 15, 1929–1958. doi:

10.1214/12-AOS1000.

Thibos, L. N. et al. (2004a) Accuracy and precision of

objective refraction from wavefront aberrations,

Journal of Vision, 4(4), pp. 329–351. doi:

10:1167/4.4.9.

Thibos, L. N., Bradley, A. and Hong, X. (2002) A statistical

model of the aberration structure of normal, well-

HEALTHINF 2019 - 12th International Conference on Health Informatics

204

corrected eyes, Ophthalmic Physiol Opt, 22(5), 427–

433.

Thibos, L. N., Wheeler, W. and Horner, D. (1997) Power

vectors: an application of Fourier analysis to the

description and statistical analysis of refractive error,

Optom Vis Sci. 1997/06/01, 74(6), 367–375.

Wang, Y. et al. (2003) Changes of higher order aberration

with various pupil sizes in the myopic eye, Journal of

refractive surgery, 19(2), 270-274. doi: 10.3928/1081-

597X-20030302-21.

Zadnik, K., Mutti, D. O. and Adams, A. J. (1992) The

repeatability of measurement of the ocular components,

Invest Ophthalmol Vis Sci, 33(7), 2325–2333.

Machine Learning based Predictions of Subjective Refractive Errors of the Human Eye

205