=Paper=
{{Paper
|id=Vol-1638/Paper6
|storemode=property
|title=Evaluation of aberrations in the optical system of the human eye based on the spatial spectrum of a diagnostic image
|pdfUrl=https://ceur-ws.org/Vol-1638/Paper6.pdf
|volume=Vol-1638
|authors=Nataly Yu. Ilyasova,Dmitriy A. Abulkhanov,Alexander V. Kupriyanov,Alexey V. Karsakov
}}
==Evaluation of aberrations in the optical system of the human eye based on the spatial spectrum of a diagnostic image ==
https://ceur-ws.org/Vol-1638/Paper6.pdf
Computer Optics and Nanophotonics
EVALUATION OF ABERRATIONS IN THE OPTICAL
SYSTEM OF THE HUMAN EYE BASED ON THE
SPATIAL SPECTRUM OF A DIAGNOSTIC IMAGE
N.Yu. Ilyasova1,2, D.А. Abulkhanov2, А.V. Kupriyanov1,2, A.V. Karsakov2
1
Image Processing Systems Institute - Branch of the Federal Scientific Research Centre “Crys-
tallography and Photonics” of Russian Academy of Sciences, Samara, Russia
2
Samara National Research University, Samara, Russia
Abstract. This study deals with the analysis of wavefront aberrations that occur
in the image owing to curvature variations in the human eye cornea. The analy-
sis is based on the Liou-Brennan human eye model. The surface curvature and
wavefront aberrations are described using Zernike polynomials. Our approach
involves evaluating parameters of the distorting system's transfer function and
measuring the expansion coefficients based on pattern recognition methods.
Constructing a feature space of medical images enables the evaluation of aber-
rations in the eye optical system to be performed.
Keywords: aberrations in the human eye, Zernike polynomials, image spectral
analysis.
Citation: Ilyasova NYu, Abulkhanov DА, Kupriyanov AV, Karsakov AV.
Evaluation of aberrations in the optical system of the human eye based on the
spatial spectrum of a diagnostic image. CEUR Workshop Proceedings, 2016;
1638: 39-48. DOI: 10.18287/1613-0073-2016-1638-39-48
1 Introduction
Our study aims to address a problem of the differential diagnosis of the human eye
optical structures through analyzing higher-order aberrations of the focusing system.
The human eye has been known to provide the major proportion of information about
the world [1-4]. Impaired eye-vision deteriorates the quality of life. The immense
experience science has gained in the treatment of eye diseases indicates that a key
culprit is age-related changes in the eye optics, which can be evaluated based on wave
aberrations. Higher-order (higher than astigmatism-related) aberrations cannot be
corrected for and detected by standard optical techniques, while patients may com-
plain of an impaired vision. At the same time, the eye may be affected by optical-
system-unrelated diseases which can also have a potentially adverse effect on the
vision. If higher-order aberrations and nonoptical abnormalities occur simultaneously
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Computer Optics and Nanophotonics Ilyasova NYu, Abulkhanov DА. et al…
it becomes impossible to make a differential diagnosis in practice in each individual
case. Measurements of optical aberrations in the human eye are also of great im-
portance in eye surgery. The possibility to detect optical aberrations during a routine
check-up would enable clinical data to be adequately interpreted, while reducing the
number of errors in choosing treatment tactics.
To address the above-described problems, a variety of optical models of the human
eye have been proposed [5-7], which include dozens of parameters of refractive sur-
faces and eye tissues. Most often encountered in the literature is the eye model devel-
oped by Liou and Brennan in 1997 [5]. As a key innovation, this model takes into
account the gradient refractive index of the crystalline lens. Currently, optical re-
search has been under way aimed at improving the existing human eye models via
introducing special aspherics on different surfaces and accounting for modern bio-
metric measurements. One of key objectives is selecting eye model parameters based
on the data of ophthalmological examination of an eye [7]. Based on medical images,
the quality of an image formed in the eye can be evaluated, the disease diagnosed, and
a treatment mode recommended.
2 Image construction model
In fig. 1 the conditional diagram of an eye explaining the course of rays in case of
image generation is shown.
Fig. 1. Scheme eye image forming: 1 – cornea, 2 – crystalline lens, 3 – iris, 4 – vitreous body,
5 – retina, 6 – optic nerve
In Ref. [7] it was studied in Zemax in which way distortions of various refracting
surfaces affected the quality of an imaging system of the human eye. The study was
based on the Liou-Brennan human eye model with a gradient crystalline lens (Fig 2)
[5].
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Computer Optics and Nanophotonics Ilyasova NYu, Abulkhanov DА. et al…
Fig. 2. A schematic layout of the refracting surfaces of a human eye according to the Liou-
Brennan model: 1– front corneal surface, 2– rear corneal surface, 3 – iris with a pupil, 4 – front
crystalline lens surface, 5– interface between the crystalline lens' composite surfaces, 6– rear
crystalline lens surface, and 7– retina.
The shape of the front cornea surface was simulated using a relationship composed of
Zernike polynomials [8]:
cr 2 N
z r, Ai Z i , ,
1 1 1 k c 2 r 2 i 1
where c is the surface curvature (reciprocal of radius), k is a conical constant (
k 2 , where is the eccentricity, with a hyperboloid of rotation being observed at
k 1 , a paraboloid at k 1 , an ellipsoid at 1 k 0, or k 0 , and a sphere at
k 0 ), N is the number of Zernike polynomials, Z i , is the i-th Zernike poly-
nomial in (2), Ai is the weight of the i-th Zernike polynomial, r is the radial coordi-
nate, is the angular coordinate, = r / R is the normalized ray coordinate, and R is
the normalization radius.
The Zernike polynomials are expressed as follows [9, 10]. An even polynomial:
Z nm , Rnm cos m , an odd polynomial: Z n m , Rnm sin m , where
nm
1 n k !
k
2
R m
n 2 k .
nm nm
n
k 0
k ! k ! k !
2 2
The wavefront aberrations can be expressed through Zernike polynomials [6]. Thus,
Defocusing: Z20 (, ) (22 1) ,
Third-order spherical aberration:
Z40 (, ) (64 62 1) ,
Distortion: Z11 (, ) cos ,
Coma: Z31 (, ) (33 2) cos ,
Astigmatism Z22 (, ) 2 cos 2 .
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Computer Optics and Nanophotonics Ilyasova NYu, Abulkhanov DА. et al…
An output image of the diffractive aberration analyzer is composed of correlation
peaks with their intensities being proportional to the contribution of the corresponding
Zernike polynomial (and related aberration) to the wavefront under analysis. The
angular dependence of the Zernike polynomials can be expressed via either cosine-
sine or exponential functions. In the latter case, wavefront aberrations can be detected
irrespective of the angular rotation.
3 Evaluating aberrations
To obtain diagnostically significant information relating to aberration parameters of
the human eye optical system, coefficients of the expansion in terms of Zernike poly-
nomials need to be derived using digital image processing techniques.
Characterizing the root-mean-square deviation of the real wavefront from an ideal
one, Zernike coefficients are used for obtaining quantitative characteristics of the
optical image quality. The lower-order coefficients describe optical aberrations: defo-
cusing (ametropia) and astigmatism. The higher-order coefficients define spherical
aberration of slanted rays incident at an angle to the eye's optical axis. They stem
from optical element asymmetry, leading to a shift of the cornea center relative to the
crystalline lens and foveola center. Another contributing factor is a higher refractive
power of the crystalline lens periphery when compared with its center for parallel
incident rays. For known aberrations, it is possible to calculate the Strehl factor (de-
fined as the ratio of the peak aberrated image intensity from a point source for the eye
under analysis to the maximum intensity for an aberration-free eye) which correlates
well with visual acuity under certain conditions. Based on the value of the Stehl fac-
tor, it becomes possible to prognosticate the vision acuity by simulating the imaging
of any optotypes on the retina. In this work we aim to study in which way aberrations
in the eye's focusing system affect the focal spot generated in the retina, as well as
identifying the relationship between the eye condition diagnosed and the aberration
coefficients measured using digital image processing techniques. We utilize machine
learning methods using Zemax model images (Fig. 2), which simulate distortions
associated with different corneal refractive surfaces in the Liou-Brennan eye model
with a GRIN crystalline lens and are described by a particular Zernike polynomial. In
the course of imaging simulation, the curvature of the front elliptical corneal surface
is varied by introducing different Zernike polynomials with different weights into its
components. This corresponds to the superposition of aberrations with weight coeffi-
cients that describe age-related changes and pathologies of the eye. Our idea is to
evaluate the parameters of a transfer function of the distorting system and apply pat-
tern recognition methods to measuring the coefficients of interest. To these ends, a
feature-based description of distortion transforms needs to be formed using model data.
Also, a classifier needs to be constructed using which it would be possible to identify
aberrations, enabling the disease to be diagnosed.
The model data is in the form of gray-level 8-bit graphical images, derived via dis-
cretization of the output wavefront intensity in the eye model (Fig. 2).
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Computer Optics and Nanophotonics Ilyasova NYu, Abulkhanov DА. et al…
а) b) c)
Fig. 3. а) Original image and b,c) images distorted using different transforms.
To obtain a discrete spectrum of the initial and distorted images, let us preliminarily
complement the 1024x1024 images with zeros, thus, obtaining 4096x4096 images
(Fig. 4). As a characteristic of the distortion transform type, below we use a space of
features derived by analyzing the transformation system's frequency response. In a
matrix form, it can be written as Hi , j Yi , j X i , j where Y and X are, respectively,
matrices of a Fourier transform of the distorted and initial images, with the inversion
and multiplication performed element-wise.
Figure 5 shows that owing to good discernibility of the image, the feature space can
be in the form of an image scan of the amplitude of the frequency response's central
part for the distortions under analysis viewed at different angles.
а) b) c)
Fig. 4. Amplitude of the central part of the Fourier transform of the images in Fig. 3.
Table 1 illustrates fragments of the frequency response amplitude images, which
show good discernibility at different weights corresponding to Zernike polynomials.
We propose using a method of principal components [11] enabling the feature space
to be formed based on eigenvectors.
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Computer Optics and Nanophotonics Ilyasova NYu, Abulkhanov DА. et al…
Fig. 5. Distorted images for different transformations, which are characterized by a lower con-
tribution of the Zernike polynomial into the cornea curvature, the respective amplitude and
phase of the frequency response's central part for a distortion transform and profiles of the
frequency response amplitude taken at different angles ( Z 2 , , weight A = 0,1).
2
4 Method of principal components
For feature extracting we use Eigenfaces method [11] which produces dimension
reduction by allowing the smaller set of basis images to represent the original training
images. Eigenfaces are extracted out of the image data by means of principal compo-
nent analysis (PCA) in the following manner:
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Fig. 6. Distorted images for different transformations, which are characterized by a higher
contribution of the Zernike polynomial into the cornea curvature, the respective amplitude and
phase of the frequency response's central part for a distortion transform and profiles of the
frequency response amplitude taken at different angles ( Z 2 , , weight A = 0,2).
2
1. Given N images with the size of h w , each image is transformed into a vector of
size m hw and placed into the set {J1 , J 2 , , J N } .
2. Each vector differs from the average by the vector i J i M , where the average
1 N
vector is defined by M Ji .
N i 1
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Computer Optics and Nanophotonics Ilyasova NYu, Abulkhanov DА. et al…
1 N
3. The covariance matrix 𝐂 ∈ 𝐑𝑚×𝑚 is defined as C
N i 1
i •i .
4. Compute the eigenvalues i and eigenvectors vi of C . We use SVD decomposi-
tion.
5. Order the eigenvectors descending by their eigenvalue. The eigenfaces (or eigen-
aberrations) are the eigenvectors corresponding to the largest eigenvalues.
Table 2 a) shows examples of the eigenvectors ('eigenfaces'), which can be termed as
'eigen-aberrations' and were derived based on the principal components corresponding
to the maximum-value weight coefficients in the expansion. Table 2 b) depicts expan-
sions of the test images in terms of the eigen-aberrations in Table 2 a).
Table 1. Exemplary images of the frequency response amplitude of the fragments under analy-
sis obtained for a) small weight and b) large weight.
Z00 , Z11 , Z 22 ,
a)
b)
Summing up, we have shown that coefficients of an expansion in terms of eigen-
aberrations can be used as a feature space. A future study will be aimed at construct-
ing and analyzing a classifier with use of the feature space proposed here.
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Computer Optics and Nanophotonics Ilyasova NYu, Abulkhanov DА. et al…
Table 2. (а) Examples of eigen-aberrations and (b) expansions in terms of the above eigen-
aberrations corresponding to maximum expansion coefficients.
Eigen-aberrations
Е1 Е2 Е3
a)
Expansions of the test images in terms of eigen-aberrations
b)
Expansion coefficients
Е1 9.3927 2.9947 6.5108
Е2 1.8091 9.0778 1.2826
Е3 1.3008 0.3778 2.3879
5 Conclusion
We have proposed a method for analyzing wavefront aberrations that occur in the
image due to variations of the cornea surface curvature in the human eye. The analy-
sis was based on a Liou-Brennan eye model. The surface curvature and wavefront
aberrations were described in this work using Zernike polynomials. We have pro-
posed that the transfer function parameters of a distorting system should be evaluated
and pattern recognition methods used for measuring the expansion coefficients. A
feature space of medical images has been constructed, enabling aberrations in the
optical system of an eye to be evaluated.
Acknowledgements
The work has been performed with partial financial support from the Ministry of Edu-
cation and Sciences of the Russian Federation within the framework of implementa-
Information Technology and Nanotechnology (ITNT-2016) 47
Computer Optics and Nanophotonics Ilyasova NYu, Abulkhanov DА. et al…
tion of the Program for Improving the SSAU Competitiveness among the World's
Leading Research and Educational Centers for the Period of 2013-2020s; under the
RFBR grants (the Russian Foundation for Basic Researches) 14-07-97040, 15-29-
03823, 15-29-07077, 16-57-48006; within the Basic Research Program No. 6 ONIT
RAN of the Russian Academy of Sciences “Mathematical Methods and Information
Technologies for the Analysis of Biomedical Images in Medical Diagnostics Applica-
tions” 2016.
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