Reference numberISO 24157:2008(E)©ISO 2008INTERNATIONAL STANDARD ISO24157First edition2008-07-01Ophthalmic optics and instruments — Reporting aberrations of the human eye Optique et instruments ophtalmiques — Méthodes de présentation des aberrations de l œil humain Copyright International Organization for Standardization Provided by IHS under license with ISO Not for ResaleNo reproduction or networking permitted without license from IHS--`,,```,,,,````-`-`,,`,,`,`,,`---ISO 24157:2008(E) PDF disclaimer This PDF file may contain embedded typefaces. In accordance with Adobe s licensing policy, this file may be printed or viewed but shall not be edited unless the typefaces which are embedded are licensed to and installed on the computer performing the editing. In downloading this file, parties accept therein the responsibility of not infringing Adobe s licensing policy. The ISO Central Secretariat accepts no liability in this area. Adobe is a trademark of Adobe Systems Incorporated. Details of the software products used to create this PDF file can be found in the General Info relative to the file; the PDF-creation parameters were optimized for printing. Every care has been taken to ensure that the file is suitable for use by ISO member bodies. In the unlikely event that a problem relating to it is found, please inform the Central Secretariat at the address given below. COPYRIGHT PROTECTED DOCUMENT © ISO 2008 All rights reserved. Unless otherwise specified, no part of this publication may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying and microfilm, without permission in writing from either ISO at the address below or ISO s member body in the country of the requester. ISO copyright office Case postale 56 • CH-1211 Geneva 20 Tel. + 41 22 749 01 11 Fax + 41 22 749 09 47 E-mail

[email protected]so.org Web www.iso.org Published in Switzerland ii © ISO 2008 – All rights reservedCopyright International Organization for Standardization Provided by IHS under license with ISO Not for ResaleNo reproduction or networking permitted without license from IHS--`,,```,,,,````-`-`,,`,,`,`,,`---ISO 24157:2008(E) © ISO 2008 – All rights reserved iiiContents 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 Copyright International Organization for Standardization Provided by IHS under license with ISO Not for ResaleNo reproduction or networking permitted without license from IHS--`,,```,,,,````-`-`,,`,,`,`,,`---ISO 24157:2008(E) iv © ISO 2008 – All rights reservedForeword 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. Copyright International Organization for Standardization Provided by IHS under license with ISO Not for ResaleNo reproduction or networking permitted without license from IHS--`,,```,,,,````-`-`,,`,,`,`,,`---INTERNATIONAL STANDARD ISO 24157:2008(E)© ISO 2008 – All rights reserved 1Ophthalmic 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 () ( )mmmnnnZ NR Mmρ θ= (1) where mnN is the normalization term; mnR is the radial term; 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. Copyright International Organization for Standardization Provided by IHS under license with ISO Not for ResaleNo reproduction or networking permitted without license from IHS--`,,```,,,,````-`-`,,`,,`,`,,`---ISO 24157:2008(E) 2 © ISO 2008 – All rights reserved3.2.1 radial term Zernike polynomial function term with indices n and m given by the equation ()()( )() ()( )0,5201!!0,5 ! 0,5 !nm smnsnsnsRsnms nmsρρ−−=−−=⎡⎤⎡⎤+− −−⎣⎦⎣⎦∑(2) 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 raρ = (3) 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 ( ) ( )cosM mmθ θ= if m W 0 (4) () ()sinMm mθ θ= if m =∑(11) 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 Copyright International Organization for Standardization Provided by IHS under license with ISO Not for ResaleNo reproduction or networking permitted without license from IHS--`,,```,,,,````-`-`,,`,,`,`,,`---ISO 24157:2008(E) © ISO 2008 – All rights reserved 53.8 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: ( )all and ,(, )mm nnnmWxyZ xycxx∂∂=∂∂∑and ( )all and ,(, )mm nnnmWxyZ xycyy∂∂=∂∂∑(12) NOTE Measured gradient values are referred to by βx(x,y) and βy(x,y) at locations x,y. Table 1 — Symbols Symbol Name Definition given in(),Amθ α meridional term for magnitude/axis Zernike functions 5.1.9 mnc Zernike coefficient 3.3 nmc Zernike coefficient – magnitude 5.1.9 m meridional index for Zernike functions 3.2 ()mnM mθ meridional term for Zernike functions 3.2.3 n radial index for Zernike functions 3.2 mnN normalization term for Zernike functions 3.2.5 ()mnR ρ radial term for Zernike functions 3.2.1 mnZ Zernike function [alternate notation: Z(n,m)] 3.2 nmZ Zernike function – magnitude/axis form 5.1.9 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,yβfitgradient fit error 5.3 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. Copyright International Organization for Standardization Provided by IHS under license with ISO Not for ResaleNo reproduction or networking permitted without license from IHS--`,,```,,,,````-`-`,,`,,`,`,,`---ISO 24157:2008(E) 6 © ISO 2008 – All rights reservedThe 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 polynomial function shall be designated by the formmnZ . 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. Copyright International Organization for Standardization Provided by IHS under license with ISO Not for ResaleNo reproduction or networking permitted without license from IHS--`,,```,,,,````-`-`,,`,,`,`,,`---ISO 24157:2008(E) © ISO 2008 – All rights reserved 75.1.6 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 index of the function, n. Therefore, a Zernike coefficient shall be designated by the formmnc . 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 00Z piston 11Z−vertical tilt11Z horizontal tilt 22Z−oblique astigmatism 02Z myopic defocus (positive coefficient value) hyperopic defocus (negative coefficient value) 22Z against the rule astigmatism (positive coefficient value) with the rule astigmatism (negative coefficient value) 33Z−oblique trefoil 13Z−vertical coma – superior steepening (positive coefficient value) vertical coma – inferior steepening (negative coefficient value) 13Z horizontal coma 33Z horizontal trefoil 44Z−oblique quatrefoil 24Z−oblique secondary astigmatism 04Z spherical aberration positive coefficient value – pupil periphery more myopic than centre negative coefficient value – pupil