EN ISO 24157:2008
(Main)Ophthalmic optics and instruments - Reporting aberrations of the human eye (ISO 24157:2008)
Ophthalmic optics and instruments - Reporting aberrations of the human eye (ISO 24157:2008)
ISO 24157:2008 specifies standardized methods for reporting aberrations of the human eye.
Augenoptik und ophthalmische Instrumente - Verfahren zur Darstellung von Abbildungsfehlern des menschlichen Auges (ISO 24157:2008)
Diese Internationale Norm legt genormte Verfahren zur Darstellung der Aberrationen des menschlichen Auges fest.
Optique et instruments ophtalmiques - Méthodes de présentation des aberrations de l'oeil humain (ISO 24157:2008)
L'ISO 24157:2008 spécifie les méthodes normalisées permettant de consigner les aberrations de l'œil humain.
Očesna optika in instrumenti - Postopek prikaza aberacije človeškega očesa (ISO 24157:2008)
General Information
Relations
Standards Content (Sample)
SLOVENSKI STANDARD
01-september-2008
2þHVQDRSWLNDLQLQVWUXPHQWL3RVWRSHNSULND]DDEHUDFLMHþORYHãNHJDRþHVD ,62
Ophthalmic optics and instruments - Reporting aberrations of the human eye (ISO
24157:2008)
Augenoptik und ophthalmische Instrumente - Verfahren zur Darstellung von
Abbildungsfehlern des menschlichen Auges (ISO 24157:2008)
Optique et instruments ophtalmiques - Méthodes de présentation des aberrations de
l'oeil humain (ISO 24157:2008)
Ta slovenski standard je istoveten z: EN ISO 24157:2008
ICS:
11.040.70 Oftalmološka oprema Ophthalmic equipment
2003-01.Slovenski inštitut za standardizacijo. Razmnoževanje celote ali delov tega standarda ni dovoljeno.
EUROPEAN STANDARD
EN ISO 24157
NORME EUROPÉENNE
EUROPÄISCHE NORM
July 2008
ICS 11.040.70
English Version
Ophthalmic optics and instruments - Reporting aberrations of
the human eye (ISO 24157:2008)
Optique et instruments ophtalmiques - Méthodes de Augenoptik und ophthalmische Instrumente - Verfahren zur
présentation des aberrations de l'oeil humain (ISO Darstellung von Abbildungsfehlern des menschlichen
24157:2008) Auges (ISO 24157:2008)
This European Standard was approved by CEN on 29 May 2008.
CEN members are bound to comply with the CEN/CENELEC Internal Regulations which stipulate the conditions for giving this European
Standard the status of a national standard without any alteration. Up-to-date lists and bibliographical references concerning such national
standards may be obtained on application to the CEN Management Centre or to any CEN member.
This European Standard exists in three official versions (English, French, German). A version in any other language made by translation
under the responsibility of a CEN member into its own language and notified to the CEN Management Centre has the same status as the
official versions.
CEN members are the national standards bodies of Austria, Belgium, Bulgaria, Cyprus, Czech Republic, Denmark, Estonia, Finland,
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Romania, Slovakia, Slovenia, Spain, Sweden, Switzerland and United Kingdom.
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© 2008 CEN All rights of exploitation in any form and by any means reserved Ref. No. EN ISO 24157:2008: E
worldwide for CEN national Members.
Contents Page
Foreword.3
Foreword
This document (EN ISO 24157:2008) has been prepared by Technical Committee ISO/TC 172 "Optics and
optical instruments" in collaboration with Technical Committee CEN/TC 170 “Ophthalmic optics” the
secretariat of which is held by DIN.
This European Standard shall be given the status of a national standard, either by publication of an identical
text or by endorsement, at the latest by January 2009, and conflicting national standards shall be withdrawn at
the latest by January 2009.
Attention is drawn to the possibility that some of the elements of this document may be the subject of patent
rights. CEN [and/or CENELEC] shall not be held responsible for identifying any or all such patent rights.
According to the CEN/CENELEC Internal Regulations, the national standards organizations of the following
countries are bound to implement this European Standard: Austria, Belgium, Bulgaria, Cyprus, Czech
Republic, Denmark, Estonia, Finland, France, Germany, Greece, Hungary, Iceland, Ireland, Italy, Latvia,
Lithuania, Luxembourg, Malta, Netherlands, Norway, Poland, Portugal, Romania, Slovakia, Slovenia, Spain,
Sweden, Switzerland and the United Kingdom.
Endorsement notice
The text of ISO 24157:2008 has been approved by CEN as a EN ISO 24157:2008 without any modification.
INTERNATIONAL ISO
STANDARD 24157
First edition
2008-07-01
Ophthalmic optics and instruments —
Reporting aberrations of the human eye
Optique et instruments ophtalmiques — Méthodes de présentation des
aberrations de l'œil humain
Reference number
ISO 24157:2008(E)
©
ISO 2008
ISO 24157:2008(E)
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Published in Switzerland
ii © ISO 2008 – All rights reserved
ISO 24157:2008(E)
Contents Page
Foreword. iv
1 Scope . 1
2 Normative references . 1
3 Terms and definitions. 1
4 Coordinate system. 5
5 Representation of wavefront data. 6
5.1 Representation of wavefront data with the use of Zernike polynomial function coefficients. 6
5.2 Representation of wavefront data in the form of wavefront gradient fields or wavefront
error function values . 9
5.3 Gradient fit error . 10
6 Presentation of data representing the aberrations of the human eye . 10
6.1 General. 10
6.2 Aberration data presented in the form of normalized Zernike coefficients. 11
6.3 Aberration data presented in the form of normalized Zernike coefficients given in
magnitude/axis form. 11
6.4 Aberration data presented in the form of topographical maps . 12
6.5 Presentation of pooled aberration data. 14
Annex A (informative) Methods of generating Zernike coefficients . 15
Annex B (informative) Conversion of Zernike coefficients to account for differing aperture sizes,
decentration and coordinate system rotation . 17
Annex C (informative) Conversion between Zernike coefficients represented in different systems
of notation . 25
Annex D (informative) Computer algorithm to generate partial derivative weighting matrices for
un-normalized Zernike polynomial functions . 27
Annex E (informative) Table of normalized Zernike polynomial functions (to 6th radial order). 29
Bibliography . 31
ISO 24157:2008(E)
Foreword
ISO (the International Organization for Standardization) is a worldwide federation of national standards bodies
(ISO member bodies). The work of preparing International Standards is normally carried out through ISO
technical committees. Each member body interested in a subject for which a technical committee has been
established has the right to be represented on that committee. International organizations, governmental and
non-governmental, in liaison with ISO, also take part in the work. ISO collaborates closely with the
International Electrotechnical Commission (IEC) on all matters of electrotechnical standardization.
International Standards are drafted in accordance with the rules given in the ISO/IEC Directives, Part 2.
The main task of technical committees is to prepare International Standards. Draft International Standards
adopted by the technical committees are circulated to the member bodies for voting. Publication as an
International Standard requires approval by at least 75 % of the member bodies casting a vote.
Attention is drawn to the possibility that some of the elements of this document may be the subject of patent
rights. ISO shall not be held responsible for identifying any or all such patent rights.
ISO 24157 was prepared by Technical Committee ISO/TC 172, Optics and photonics, Subcommittee SC 7,
Ophthalmic optics and instruments.
iv © ISO 2008 – All rights reserved
INTERNATIONAL STANDARD ISO 24157:2008(E)
Ophthalmic optics and instruments — Reporting aberrations of
the human eye
1 Scope
This International Standard specifies standardized methods for reporting aberrations of the human eye.
2 Normative references
The following referenced documents are indispensable for the application of this document. For dated
references, only the edition cited applies. For undated references, the latest edition of the referenced
document (including any amendments) applies.
ISO 8429, Optics and optical instruments — Ophthalmology — Graduated dial scale
3 Terms and definitions
For the purposes of this document, the following terms and definitions apply. Symbols used are summarized
in Table 1.
3.1
line of sight
line from the point of interest in object space to the centre of the entrance pupil of the eye and continuing from
the centre of the exit pupil to the retinal point of fixation (generally the foveola)
3.2
Zernike polynomial function
one of a complete set of functions defined and orthogonal over the unit circle, the product of three terms, a
normalization term, a radial term and a meridional term, parameterized by a dimensionless radial parameter, ρ,
and a dimensionless meridional parameter, θ, designated by a non-negative radial integer index, n, and a
signed meridional index, m, and given by the equation
m
mm
Z =NR ρ M mθ (1)
() ( )
nnn
where
m
N is the normalization term;
n
m
R is the radial term;
n
M(mθ) is the meridional term;
the parameter ρ is a real number continuous over its range of 0 to 1,0;
the parameter θ is a real number continuous over its range of 0 to 2π.
NOTE For a given value of radial index n, the meridional index m may only take the values −n, −n+2, …, n−2 and n.
ISO 24157:2008(E)
3.2.1
radial term
Zernike polynomial function term with indices n and m given by the equation
0,5nm−
( ) s
()−−1!(ns)
m
ns−2
Rρρ= (2)
()
n ∑
⎡⎤⎡ ⎤
sn!0,5 +−m s ! 0,5n−m−s !
() ()
s=0
⎣⎦⎣ ⎦
where s is an integer summation index incremented by one unit
3.2.2
radial parameter
ρ
dimensionless number taking values between 0 and 1, its value at any radial distance, r, from the aperture
centre being given by the expression
r
ρ = (3)
a
where a is the value of the aperture radius
3.2.3
meridional term
Zernike polynomial function term with index m given by the equations
Mmmθ = cos θ if m W 0 (4)
( ) ( )
Mm()θ = sin m θ if m < 0 (5)
()
NOTE The meridional term is also known as the azimuthal term.
3.2.4
meridional parameter
θ
angular value taking values between 0 and 2π (0° and 360°), expressed in the coordinate system defined in
Clause 4
NOTE This is also called the azimuthal angle.
3.2.5
normalization term
Zernike polynomial function term with indices n and m, equal to 1,0 for “un-normalized” functions (3.2.7) and
for “normalized” functions (3.2.6) by the equation
m
Nn=−21δ + (6)
()
()
nm0,
where δ = 1 if m = 0, δ = 0 if m ≠ 0.
0,m 0,m
3.2.6
normalized Zernike polynomial function
Zernike polynomial function whose normalization term takes the form given in 3.2.5 for “normalized” functions
defined as orthogonal in the sense that it satisfies the following equation
12π
mm′
ρρdZZdθ =πδ δ (7)
′′′
nn n,,n mm
∫∫
2 © ISO 2008 – All rights reserved
ISO 24157:2008(E)
where
δ = 1 if n = n′, δ = 0 if n ≠ n′;
n,n′ n,n′
δ = 1 if m = m′, δ = 0 if m ≠ m′.
m,m′ m,m′
3.2.7
un-normalized Zernike polynomial function
Zernike polynomial function whose normalization term is equal to 1,0 and defined as orthogonal in the sense
that it satisfies the equation
12π
mm′
21−+δρndρZZdθ=πδδ (8)
()()
′′′
0,mnnn,nm,m
∫∫
where
δ = 1 if n = n′, δ = 0 if n ≠ n′;
n,n′ n,n′
δ = 1 if m = m′, δ = 0 if m ≠ m′;
m,m′ m,m′
δ = 1 if m = 0, δ = 0 if m ≠ 0.
0,m 0,m
3.2.8
order
value of the radial index n of a Zernike polynomial function
3.3
Zernike coefficient
m
member of a set of real numbers, c , which is multiplied by its associated Zernike function to yield a term that
n
is subsequently used in a sum of terms to give a value equal to the best estimate of the surface, S(ρ,θ), that
has been fitted with Zernike terms, such a sum being represented by
mm
Scρθ, = Z (9)
()
∑ nn
all nmand
NOTE 1 Each set of Zernike coefficients has associated with it the aperture diameter that was used to generate the set
from surface elevation data. The set is incomplete without this aperture information.
NOTE 2 Annex A gives information on a method to find Zernike coefficients from wavefront slope (gradient) data.
3.3.1
normalized Zernike coefficient
Zernike coefficient generated using normalized Zernike functions and so designed to be used with them to
reconstruct a surface
NOTE Normalized Zernike coefficients have dimensional units of length.
3.3.2
un-normalized Zernike coefficient
Zernike coefficient generated using un-normalized Zernike functions and so designed to be used with them to
reconstruct a surface
NOTE Un-normalized Zernike coefficients have dimensional units of length.
ISO 24157:2008(E)
3.4
wavefront error (of an eye)
W(x,y) or W(r,θ)
optical path-length (i.e. physical distance times refractive index) between a plane wavefront in the eye’s
entrance pupil and the wavefront of light exiting the eye from a point source on the retina, and specified as a
function (wavefront error function) of the (x,y) (or r,θ) coordinates of the entrance pupil
NOTE 1 Wavefront error is measured in an axial direction (i.e. parallel to the z-axis defined in Clause 4) from the pupil
plane towards the wavefront.
NOTE 2 By convention, the wavefront error is set to zero at the pupil centre by subtracting the central value from
values at all other pupil locations.
NOTE 3 Wavefront error has physical units of metres (typically reported in micrometres) and pertains to a specified
wavelength.
3.5
optical path-length difference
OPD
negative of the wavefront error (3.4) at each point in a wavefront representing the correction of the optical
path-length needed to correct the wavefront error
3.6
root mean square wavefront error
RMS wavefront error
〈of an eye〉 quantity computed as the square root of the variance of the wavefront error (3.4) function and
defined as
⎡⎤Wx(,y) dxdy
⎣⎦
∫∫
pupil
(10)
RMS =
WFE
A
where A is the area of the pupil
or, if the wavefront error function is expressed in terms of normalized Zernike coefficients, a quantity equal to
the square root of the sum of the squares of the coefficients with radial indices 2 or greater
m
RMS = c (11)
WFE ∑ ()n
nm>1,all
NOTE 1 Piston and average tilt should be excluded from this calculation because they correspond to lateral
displacements of the image rather than image degradation per se.
NOTE 2 The RMS error can also be found using the discrete set of wavefront error values that were used to generate
the Zernike coefficients and standard statistical methods. When this is done it might be found that this RMS value does not
exactly match the value found using the formula given above. This is more likely to happen in cases where the locations in
the pupil used to sample the wavefront error form a non-uniformly spaced grid. Then the data set does not lead to the
formation of discrete, orthogonal Zernike functions.
3.7
higher-order aberrations
those aberrations experienced by the eye in addition to sphero-cylindrical refractive errors and prismatic error
and thus, if the wavefront error is expressed in terms of Zernike polynomial function coefficients, those of
order 3 and higher
4 © ISO 2008 – All rights reserved
ISO 24157:2008(E)
3.8
wavefront gradient
∂W(x,y)
vector giving the values of the gradient of the wavefront, ∂W(x,y)/∂x and ∂W(x,y)/∂y, at locations x and y and,
when expressed in terms of Zernike polynomial coefficients, given by:
m m
∂Wx,y ∂Wx,y
( ) ∂Z (,xy) ( ) ∂Z (,xy)
m m
n n
= c and = c (12)
∑ n ∑ n
∂∂xx ∂∂yy
all nmand all nmand
NOTE Measured gradient values are referred to by β (x,y) and β (x,y) at locations x,y.
x y
Table 1 — Symbols
Symbol Name Definition given in
Amθ,α meridional term for magnitude/axis Zernike functions 5.1.9
()
m
c Zernike coefficient 3.3
n
c Zernike coefficient – magnitude 5.1.9
nm
m meridional index for Zernike functions 3.2
m
M mθ meridional term for Zernike functions 3.2.3
()
n
n radial index for Zernike functions 3.2
m
N normalization term for Zernike functions 3.2.5
n
m
R ρ radial term for Zernike functions 3.2.1
()
n
m
Z Zernike function [alternate notation: Z(n,m)] 3.2
n
Z Zernike function – magnitude/axis form 5.1.9
nm
α axis parameter for magnitude/axis form Zernike functions 5.1.9
ρ radial parameter for Zernike functions 3.2.2
θ meridional parameter for Zernike functions 3.2.4
W(x,y) wavefront error 3.4
βx,y measured gradient at a location x,y 3.8
∂Wx,y wavefront gradient at a location x,y 3.8
β gradient fit error 5.3
fit
4 Coordinate system
The coordinate system used to represent wavefront surfaces shall be the standard ophthalmic coordinate
system in accordance with ISO 8429 in which the x-axis is local horizontal with its positive sense to the right
as the examiner looks at the eye under measurement, the y-axis is local vertical with its positive sense
superior with respect to the eye under measurement, the z-axis is the line of sight of the eye under
measurement with its positive sense in the direction from the eye toward the examiner. The horizontal and
vertical origin of the coordinate system is the centre of the visible pupil of the eye. The coordinate system
origin lies in the plane of the exit pupil of the eye (for light originating on the retina and passing out through the
pupil). This coordinate system is illustrated in Figure 1.
ISO 24157:2008(E)
The sign convention used for wavefront error values reported at any location on a wavefront shall be that used
for this coordinate system.
When Zernike coefficients are used to represent a wavefront or to report wavefront error, the sign convention
used to describe the individual Zernike functions shall be that used for this coordinate system.
a) Coordinate system b) Clinician's view of patient
Key
OD right eye
OS left eye
Figure 1 — Ophthalmic coordinate system (ISO 8429)
5 Representation of wavefront data
5.1 Representation of wavefront data with the use of Zernike polynomial function
coefficients
5.1.1 Symbols for Zernike polynomial functions
Zernike polynomial functions shall be designated by the upper case letter Z followed by a superscript and a
subscript. The superscript shall be a signed integer representing the meridional index of the function, m. The
subscript shall be a non-negative integer representing the radial index of the function, n. Therefore a Zernike
m
polynomial function shall be designated by the form Z .
n
If, for reasons of font availability, it is not possible to write superscript and subscripts, the Zernike polynomial
functions may be represented as a upper case letter Z followed by parentheses in which the radial index, n,
appears first, followed, after a comma, by the meridional index, m, thus Z(n,m).
5.1.2 Radial index
The radial index shall be designated by the lower case letter n.
5.1.3 Meridional index
The meridional index shall be designated by the lower case letter m.
5.1.4 Radial parameter
The radial parameter shall be designated by the Greek letter ρ.
5.1.5 Meridional parameter
The meridional parameter shall be designated by the Greek letter θ.
6 © ISO 2008 – All rights reserved
ISO 24157:2008(E)
5.1.6 Coefficients
When a surface is represented by Zernike coefficients, these coefficients shall be designated by the lower
case letter c followed by a superscript and a subscript. The superscript shall be a signed integer representing
the meridional index of the function, m. The subscript shall be a non-negative integer representing the radial
m
index of the function, n. Therefore, a Zernike coefficient shall be designated by the form c .
n
5.1.7 Common names of Zernike polynomial functions
Zernike polynomial functions are often referred to by their common names. These names are given in Table 2
in so far as the functions have been given a common name.
Table 2 — Common names of Zernike polynomial functions
Zernike function Common name
Z piston
−1
Z vertical tilt
Z horizontal tilt
−2
Z oblique astigmatism
myopic defocus (positive coefficient value)
Z
hyperopic defocus (negative coefficient value)
against the rule astigmatism (positive coefficient value)
Z
with the rule astigmatism (negative coefficient value)
−3
Z oblique trefoil
vertical coma – superior steepening (positive coefficient value)
−1
Z
vertical coma – inferior steepening (negative coefficient value)
Z horizontal coma
Z horizontal trefoil
−4
Z oblique quatrefoil
−2
Z oblique secondary astigmatism
spherical aberration
Z positive coefficient value – pupil periphery more myopic than centre
negative coefficient value – pupil periphery more hyperopic than centre
Z with/against the rule secondary astigmatism
Z quatrefoil
−1
Z secondary vertical coma
Z secondary horizontal coma
ISO 24157:2008(E)
5.1.8 Comparison of data expressed as Zernike coefficients generated using different aperture sizes
The Zernike coefficient values describing a given wavefront error depend on the aperture size used when they
are generated from measurement data. Due to this dependence on pupil diameter, different coefficient values
will be found to describe the wavefront error of a given eye if the pupil size changes from one measurement to
the next. Therefore, to adequately compare the wavefront error of the same eye at different times or to
compare the wavefront errors of two eyes using Zernike coefficients, the compared coefficients shall have
been generated using the same pupil diameter even though measurements were taken with different pupil
diameters. Zernike coefficients taken at one pupil diameter may be converted into values for a second, smaller
pupil diameter using either the method given in Annex B or a similar method.
Wavefront error comparisons using Zernike coefficients found in accordance with this International Standard
shall be made between sets of Zernike coefficients that have be converted to a common pupil diameter.
5.1.9 Representation of wavefront error data expressed as Zernike coefficients presented in
magnitude/axis form
Zernike terms of the same radial order, n, and having meridional indices, m, with the same magnitude but with
opposite signs may be considered to represent the two components of a vector in an angular space with a
multiplicity equal to the magnitude of m. It is therefore possible to define Zernike functions that combine the
functions defined in 3.2 having the same radial order, n, and meridional indices with the same magnitude into
a new set of functions defined by
m
m
ZNρ,,θα =R ρAmθ,α (13)
() () ( )
nm n n
where
m
R ()ρ is defined by 3.2.1;
n
m
N is defined by 3.2.5;
n
⎡⎤
Am()θ,cαθ=−os m(α)
⎣⎦
and where α is an angular parameter giving the orientation of the vector in space.
A surface, S(ρ,θ), such as a wavefront error, is expressed using these Zernike functions as
Scρθ,,= Z ρθ,α
( ) ()
∑ nm nm nm
all nmand
where the coefficients c and the angular parameters α are related to the coefficients defined in 3.3 by the
nm nm
equations
−mm
cc=+c (14)
() ( )
nm n n
−m
⎛⎞
c
n
⎜⎟
a tan
m
⎜⎟
c
⎝⎠n
α = (15)
nm
m
8 © ISO 2008 – All rights reserved
ISO 24157:2008(E)
5.1.10 Common names of Zernike polynomial functions – magnitude/axis form
Zernike polynomial functions are often referred to by their common names. For the magnitude/axis Zernike
functions defined in 5.1.9, these names are given in Table 3 in so far as the functions have been given a
common name.
Table 3 — Common names of Zernike polynomial functions – magnitude/axis form
Zernike function Common name
Z piston
Z tilt
myopic defocus (positive coefficient value)
Z
hyperopic defocus (negative coefficient value)
astigmatism
Z against the rule, axis = 180°
with the rule, axis = 90°
Z coma
Z trefoil
spherical aberration
positive coefficient value – pupil periphery more myopic than
Z centre
negative coefficient value – pupil periphery more hyperopic
than centre
Z secondary astigmatism
Z quatrefoil
Z secondary coma
5.2 Representation of wavefront data in the form of wavefront gradient fields or wavefront
error function values
5.2.1 Gradient values
The measurements made of the aberrations of the eye by aberrometers are in general measurements of the
gradient of the wavefront error function. Measurements of this type may also be thought of as measurements
of the deflection of rays from an un-aberrated direction by the optical system of the eye. In the case of rays
originating at the retina and measured as they pass the exit pupil, the deflection is measured from the ray to a
ray at the same pupil location but parallel to the line of sight. In the case of rays entering the eye through its
entrance pupil, the deflection is measured from the ray to a ray that enters the eye parallel to the line of sight
and is refracted so that it intersects the retina at the point the line of sight intersects the retina. The gradient
information consists of the two-dimensional location of the measured ray in the plane of the exit pupil of the
eye and the two components of its deflection.
So that this information may be conveyed in a standardized fashion, the data for each measured ray or
location in the wavefront will consist of four numbers. The first two are the horizontal (x) and vertical (y)
coordinates of the location given as Cartesian coordinates in the coordinate system specified in Clause 4 and
expressed in millimetres. The second two numbers are the horizontal and vertical component values of the
gradient or, to state this another way, the second two numbers are the horizontal and vertical deflections of
the ray given as tangent values.
A fifth, optional, number may be included giving the quality or certainty associated with the information given
at each data location.
ISO 24157:2008(E)
5.2.2 Wavefront error values
If the aberrations of the eye consist of the values of the wavefront error function itself, then the information
needed to express this at a given location in the wavefront consists of the location and the value of the
wavefront error functions at that location.
So that this information may be conveyed in a standardized fashion, the data for each measured ray or
location in the wavefront will consist of three numbers. The first two are the horizontal (x) and vertical (y)
coordinates of the location given as Cartesian coordinates in the coordinate system specified in Clause 4 and
expressed in millimetres. The third number is the value of the wavefront error (3.4) expressed in micrometres.
A fourth, optional, number may be included giving the quality or certainty associated with the information given
at each data location.
5.3 Gradient fit error
When the occasion arises that a wavefront error function has been reconstructed from measured wavefront
gradient data and the values for the reconstructed wavefront are conveyed either in the form of gradient fields
in accordance with 5.1 or the wavefront error function itself in accordance with 5.2, this information shall be
accompanied by two additional values for each data location that give information on the quality of the fit of
the data to the reconstruction. The first such value is the difference between the measured x gradient value
and the reconstructed x gradient value. The second value is the difference between the measured y gradient
value and reconstructed y gradient value. These two values are defined as the gradient fit error and constitute
the two components of an error gradient field. These two values are not to be taken as the same as the
optional quality values allowed for in 5.1 and 5.2 as those values refer to the quality of the measured data
themselves whereas the gradient fit values refer to the quality of the fit of the data to the reconstructed
wavefront.
The gradient fit parameter β is a metric measure that can be used to identify the overall quality of the fit. It is
fit
generally the merit function that is minimized in least-squares fitting. It is defined by:
⎡⎤
⎡⎤∂∂Wx(,y) W(x,y)
ββ(,xy)−−(x,y)
∑∑⎢⎥xy⎢⎥
∂∂xy
⎣⎦
⎣⎦
xy,,x y
β=+ (16)
fit
NN
The various parameters are defined in 3.8.
6 Presentation of data representing the aberrations of the human eye
6.1 General
The preferred method of communicating aberration data for the human eye to others so that they may analyse
them as they see them is in the form of a set of gradient components for each measured location as specified
in 5.2.1. These values fully characterize the measured wavefront error of the eye and may be used to
reconstruct that wavefront error surface using any method desired.
However, wavefront data in the form of gradient components do not convey the wavefront information in a
fashion that is easily understood nor in a form that lends itself for convenient use in papers, displays and other
common forms of communication. Thus the preferred methods for presenting aberration data of the human
eye are:
a) as a list of normalized Zernike coefficients;
b) as a bar chart showing the values of the normalized Zernike coefficients;
c) in the form of a topographical map of the wavefront surface.
10 © ISO 2008 – All rights reserved
ISO 24157:2008(E)
6.2 Aberration data presented in the form of normalized Zernike coefficients
6.2.1 Aperture information
When data representing the aberrations of the human eye are presented in the form of normalized Zernike
coefficients, the aperture diameter used in generating the coefficient shall form a part of the data set and shall
be the first member of the set.
6.2.2 Units
Coefficients shall be given in units of micrometres. Aperture size shall be given in millimetres.
6.2.3 Ordering of terms
When data representing the aberrations of the human eye are presented in the form of Zernike coefficients,
the coefficients shall be listed in the following order.
The first value of the coefficient set shall be the value of the aperture diameter used to generate the set,
followed by the Zernike coefficients grouped by common radial index, n, with these groups listed in increasing
magnitude of the radial index. Within a group of common radial indices, the coefficients shall be listed in
increasing value of the meridional index, m, starting with the most negative m value and proceeding in
increasing order to the most positive m value. The following single index j gives the above ordering for the
Zernike coefficients. The first Zernike coefficient of the set has the ordering index value zero.
nn(2++) m
j = (17)
6.2.4 Form of presentation
6.2.4.1 Tabular form
When the normalized Zernike coefficients are presented as a table the first column shall contain the Zernike
function symbols ordered in accordance with 6.2.3. The second column shall contain the numerical values of
the coefficients, aligned with their respective function symbols. If it is desired to present names for the Zernike
functions, these names may be placed in a third column. The first row of the table shall contain the words
“aperture diameter” in the first column and the aperture diameter value in the second column.
6.2.4.2 Bar chart form
When the normalized Zernike coefficients are presented as a bar chart the values assigned to the bars shall
be values of the normalized Zernike coefficients ordered in accordance with 6.2.3. The bars shall be labelled
using Zernike double index symbols in accordance with 5.1.1. The values of the bars shall be given in
micrometres. The value of the aperture diameter used to create the coefficient values shall appear within the
graph.
6.3 Aberration data presented in the form of normalized Zernike coefficients given in
magnitude/axis form
6.3.1 Aperture information
When data representing the aberrations of the human eye are presented in the form of normalized Zernike
coefficients, the aperture diameter used in generating the coefficient shall form a part of the data set and shall
be the first member of the set.
ISO 24157:2008(E)
6.3.2 Units
Coefficients shall be given in units of micrometres. Aperture size shall be given in millimetres. Axes shall be
given in degrees.
6.3.3 Ordering of terms
When data representing the aberrations of the human eye are presented in the form of Zernike coefficients,
given in magnitude/axis form in accordance with 5.1.9, the coefficients, c and axis values, α , shall be
nm nm
listed in the following order.
The first value of the coefficient set shall be the value of the aperture diameter used to generate the set
followed by the coefficient magnitudes and axes grouped by common radial index, n, with these groups listed
in increasing magnitude of the radial index. Within a group of common radial indices, the coefficients shall be
listed in increasing value of the meridional index, m.
6.3.4 Tabular form of presentation
When the normalized Zernike coefficients are presented as a table the first column shall contain the Zernike
function symbols ordered in accordance with 6.3.3. The second column shall contain the numerical values of
the coefficients, aligned with their respective function symbols. The third column shall contain the numerical
values of the axes, aligned with their respective function symbols. If it is desired to present names for the
Zernike functions, these names may be placed in a fourth column. The first row of the table shall contain the
words “aperture diameter” in the first column and the aperture diameter value in the second column.
6.4 Aberration data presented in the form of topographical maps
6.4.1 General
In order to facilitate the interpretation and comparison of wavefront measurements from different systems,
criteria for a standard graphical display of ocular wavefront aberrations are established. The elements of the
display are: colour set, aberration scale, colour contour map, numeric data, spatial scale and title. If
compliance with this International Standard is claimed, this standardized display of ocular wavefront
aberrations shall be made available to the user and shall contain the text “ISO 24157”.
6.4.2 Display contents
6.4.2.1 Standardized display
This shall contain the following elements:
⎯ title;
⎯ colour legend graphic;
⎯ step size text;
⎯ colour contour map of higher-order aberrations (wavelength 0,555 µm, if possible);
⎯ numeric data;
⎯ spatial extent indicator;
⎯ axis indication;
⎯ reference to ISO 24157.
12 © ISO 2008 – All rights reserved
ISO 24157:2008(E)
6.4.2.2 Display title
The display shall be titled “Higher order aberrations”.
6.4.2.3 Colour contour map
The colour map shows a colour-coded representation of the higher-order aberrations in the entrance pupil of
the eye. If possible, these aberrations should be referenced to wavelength 0,555 µm. The centre value of the
map shall be zero corresponding to the chief ray.
6.4.2.4 Numeric data
The numeric data to be displayed on the map shall include:
⎯ low-order aberrations' RMS value, in micrometres, computed for aberrations second-order;
⎯ higher-order aberrations' RMS value, in micrometres, computed for aberrations third-order and above;
⎯ total aberrations' RMS value, in micrometres, computed for aberrations second-order and above;
⎯ diameter of pupil, in millimetres.
6.4.2.5 Spatial extent graphic
The spatial extent graphic shows the size of the colour map. It shall consist of some graphic indicator and text
indicating the width of the display in units of millimetres.
6.4.2.6 Axis indicator
The axis indicator graphic shows the angular coordinate system as defined in Clause 4.
6.4.3 Standardized scales
For display of the higher-order aberrations, the standardized display shall use one of four step size intervals:
0,1 µm, 0,2 µm, 0,5 µm, and 1,0 µm
The step size shall be prominently displayed below the colour legend. Twenty-one (21) colours shall be used
with the centre colour set at zero. If the aberration value to be displayed exceeds the range of the colour scale,
the highest or lowest colour (as appropriate) shall be used.
6.4.4 Colour palette
The twenty-one colours shall follow the general guidelines:
⎯ where the wavefront error as defined in 3.4 takes negative values, cooler colours are used (blues);
⎯ where the wavefront error as defined in 3.4 takes positive values, warmer colours are used (reds);
⎯ where the wavefront error as defined in 3.4 is zero, green is used.
ISO 24157:2008(E)
6.5 Presentation of pooled aberration data
6.5.1 General
For multiple eye studies of the aberrations of the eye it is desirable to present pooled results. When this is
done, certain precautions need to be taken both in the analysis and presentation of the data if the pooled
results are to be meaningful. In sets of data from multiple sources and from many individuals it is almost
certain that not all the data will come from eyes having the same pupil size. Therefore steps shall be taken to
account for changes in aberration values found when the pupil size changes. It is also quite likely that both
right and left eyes are included in a study. When this is the case this fact needs to be acknowledged along
with any steps that have been taken to account for known anatomical asymmetries that occur between right
and left eyes.
6.5.2 Analysis and presentation of pooled aberration results based on Zernike coefficient sets
When the data used for a study is in the form of Zernike polynomial coefficients, certain precautions need to
be taken both in the processing and analysis of the data and in its presentation. This is because the values of
the Zernike coefficients that describe a given wavefront will change, even if they are given in accordance with
this International Standard, if the aperture diameter used is changed. Therefore the coefficients may not be
directly compared until all use a common pupil diameter. For this reason the first step in the analysis of pooled
data sets is to convert all Zernike coefficient sets to sets having the same pupil size. This may be done using
the method given in Clause B.2.
If the data are given in the form specified in 5.2.1 or 5.2.2 but the pooled results are presented in the form of
Zernike coefficients, the coefficients shall be generated from the data using the same pupil diameter for all
eyes.
6.5.3 Analysis and presentation of pooled aberration data where both right and left eye data are used
When both right eye and left eye data are used in pooled data sets, consideration needs to be taken for the
known asymmetries about the vertical meridian of the eye between right and left eyes. This may be done in
one of two ways.
If aberration data include measurements from both eyes, which have not been altered to compensate for
known asymmetries, it is preferable to analyse the data and present the results separately for the right eyes
and the left eyes. If it is decided to pool data from both right and left eyes in the same analysis, this fact shall
be stated explicitly.
If it is decided to pool data from both right and left eyes in the same analysis and when analysis and
presentation is done based on Zernike coefficient sets, the known anatomical asymmetry may be accounted
for by altering the sign of all Zernike coefficients that arise from Zernike polynomials with negative, even
meridional indices, and with positive, odd meridional indices for all left eye data prior to analysis. This step has
the effect of giving left eyes the same asymmetry, on average
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