SIST 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)
Specification of standardized methods for reporting the optical aberrations of human eyes.
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
SIST EN ISO 24157:2008
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
SIST EN ISO 24157:2008 en
2003-01.Slovenski inštitut za standardizacijo. Razmnoževanje celote ali delov tega standarda ni dovoljeno.
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SIST EN ISO 24157:2008
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SIST EN ISO 24157:2008
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,
France, Germany, Greece, Hungary, Iceland, Ireland, Italy, Latvia, Lithuania, Luxembourg, Malta, Netherlands, Norway, Poland, Portugal,
Romania, Slovakia, Slovenia, Spain, Sweden, Switzerland and United Kingdom.
EUROPEAN COMMITTEE FOR STANDARDIZATION
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EUROPÄISCHES KOMITEE FÜR NORMUNG
Management Centre: rue de Stassart, 36 B-1050 Brussels
© 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.
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SIST EN ISO 24157:2008
EN ISO 24157:2008 (E)
Contents Page
Foreword.3
2
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SIST EN ISO 24157:2008
EN ISO 24157:2008 (E)
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.
3
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SIST EN ISO 24157:2008
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SIST EN ISO 24157:2008
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
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SIST EN ISO 24157:2008
ISO 24157:2008(E)
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ii © ISO 2008 – All rights reserved
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SIST EN ISO 24157:2008
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 2008 – All rights reserved iii
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SIST EN ISO 24157:2008
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
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SIST EN ISO 24157:2008
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.
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SIST EN ISO 24157:2008
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
∫∫
00
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SIST EN ISO 24157:2008
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
∫∫
00
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.
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SIST EN ISO 24157:2008
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
2
⎡⎤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
2
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
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SIST EN ISO 24157:2008
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.
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SIST EN ISO 24157:2008
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.
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SIST EN ISO 24157:2008
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
0
Z piston
0
−1
Z vertical tilt
1
1
Z horizontal tilt
1
−2
Z oblique astigmatism
2
myopic defocus (positive coefficient value)
0
Z
2
hyperopic defocus (negative coefficient value)
against the rule astigmatism (positive coefficient value)
2
Z
2
with the rule astigmatism (negative coefficient value)
−3
Z oblique trefoil
3
vertical coma – superior steepening (positive coefficient value)
−1
Z
3
vertical coma – inferior steepening (negative coefficient value)
1
Z horizontal coma
3
3
Z horizontal trefoil
3
−4
Z oblique quatrefoil
4
−2
Z oblique secondary astigmatism
4
spherical aberration
0
Z positive coefficient value – pupil periphery more myopic than centre
4
negative coefficient value – pupil periphery more hyperopic than centre
2
Z with/against the rule secondary astigmatism
4
4
Z quatrefoil
4
−1
Z secondary vertical coma
5
1
Z secondary horizontal coma
5
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SIST EN ISO 24157:2008
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
22
−mm
cc=+c (14)
() ( )
nm n n
−m
⎛⎞
c
n
⎜⎟
a tan
m
⎜⎟
c
⎝⎠n
α = (15)
nm
m
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SIST EN ISO 24157:2008
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
00
Z tilt
11
myopic defocus (positive coefficient value)
Z
20
hyperopic defocus (negative coefficient value)
astigmatism
Z against the rule, axis = 180°
22
with the rule, axis = 90°
Z coma
31
Z trefoil
33
spherical aberration
positive coefficient value – pupil periphery more myopic than
Z centre
40
negative coefficient value – pupil periphery more hyperopic
than centre
Z secondary astigmatism
42
Z quatrefoil
44
Z secondary coma
51
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
...
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