1、American National Standardfor Ophthalmics Methods of ReportingOptical Aberrations of EyesANSI Z80.28-2017ANSIZ80.28-2017ANSIZ80.28-2017Revision ofANSI Z80.28-2010American National Standardfor Ophthalmics Methods for ReportingOptical Aberrations of EyesSecretariatThe Vision CouncilApproved March 27,
2、2017Published August 21, 2017American National Standards Institute, Inc.Approval of an American National Standard requires review by ANSI that therequirements for due process, consensus, and other criteria for approval havebeen met by the standards developer.Consensus is established when, in the jud
3、gement of the ANSI Board ofStandards Review, substantial agreement has been reached by directly andmaterially affected interests. Substantial agreement means much more thana simple majority, but not necessarily unanimity. Consensus requires that allviews and objections be considered, and that a conc
4、erted effort be madetowards their resolution.The use of American National Standards is completely voluntary; theirexistence does not in any respect preclude anyone, whether he has approvedthe standards or not, from manufacturing, marketing, purchasing, or usingproducts, processes, or procedures not
5、conforming to the standards.The American National Standards Institute does not develop standards andwill in no circumstances give an interpretation of any American NationalStandard. Moreover, no person shall have the right or authority to issue aninterpretation of an American National Standard in th
6、e name of the AmericanNational Standards Institute. Requests for interpretations should beaddressed to the secretariat or sponsor whose name appears on the titlepage of this standard.CAUTION NOTICE: This American National Standard may be revised orwithdrawn at any time. The procedures of the America
7、n National StandardsInstitute require that action be taken periodically to reaffirm, revise, orwithdraw this standard. Purchasers of American National Standards mayreceive current information on all standards by calling or writing the AmericanNational Standards Institute.American National StandardPu
8、blished byThe Vision Council225 Reinekers LaneSuite 700Alexandria, VA 22314Copyright 2017 by The Vision CouncilAll rights reserved.No part of this publication may be reproduced in anyform, in an electronic retrieval system or otherwise,without prior written permission of the publisher.Developed byTh
9、e Accredited Committee Z80 for Ophthalmic Standards -The Vision CouncilZ80 Secretariat225 Reinekers LaneSuite 700Alexandria, VA 22314iContentsPageForeword ii1 Scope . 12 Normative References 13 Symbols and Definitions. 24 Coordinate system 65 Representation of wavefront data. 75.1 Representation of
10、wavefront data with the use ofZernike polynomial function coefficients. 75.2 Representation of wavefront data in the form ofwavefront gradient fields of wavefront error function values 105.3 Gradient fit error . 116 Presentation of data representing the aberrations of the human eye. 126.1 General. 1
11、26.2 Aberration data presented in the form ofnormalized Zernike coefficients 126.3 Aberration data presented in the form of normalizedZernike coefficients given in magnitude/axis form 136.4 Aberration data presented in the form of topographical maps 146.5 Presentation of pooled aberration data 16Tab
12、les1 Symbols 22 Common names of Zernike polynomial functions. 83 Common names of Zernike polynomial functions. 10Figure1 Ophthalmic coordinate system . 6AnnexesA Methods of generating Zernike coefficients 18B Conversion of Zernike coefficients to account for differingaperture sizes, decentration and
13、 coordinate system rotation. 21C Conversion between Zernike coefficients represented indifferent systems of notation. 31D Computer algorithm to generate partial derivative weighing matricesfor un-normalized Zernike polynomial functions. 33E Table of normalized Zernike polynomial functions(through 6t
14、h radial order) 35F Bibliography 37iiForeword (This foreword is not part of American National Standard ANSI Z80.28-2017.)This American National Standard addresses the communication of aberrations of theeye. With the dramatic increase in applications associated with aberrations, it is im-portant to h
15、ave uniform methods for reporting data so that it is understandable acrossapplications.ANSI Z80.28-2017 was adapted by a group of experts within the ANSI Instrumentsand Low Vision Devices Subcommittee under the chair of William L. Brown, O.D.,Ph.D. It is a performance standard.The major change made
16、in this edition of the standard is a clarification of 5.2 that 1)changes the term “wavefront gradient fields“ to “wavefront ray deflection“, and 2)places the emphasis of the clause on describing the direction of propagation of raysthat are the normals to the wavefront error surface being measured.Su
17、ggestions for improvement of this standard will be welcome. They should be sentto the Vision Council, 225 Reinekers Lane, Suite 700, Alexandria, VA 22314.This standard was processed and approved for submittal to ANSI by the AccreditedStandards Committee on Ophthalmic Optics, Z80. Committee approval
18、of this stan-dard does not necessarily imply that all committee members voted for its approval. Atthe time of approval of this standard, the Z80 Committee consisted of the followingmembers:Thomas White, M.D., ChairQuido Cappelli, Vice-ChairWilliam Benjamin, SecretaryMichael Vitale, SecretariatOrgani
19、zation Represented Name of RepresentativeAbbott Medical Optics. Leonard BormannAdvance Medical Technologies Association Michael PflegerAmerican Academy of Ophthalmology . Thomas C. WhiteAmerican Academy of Optometry. David S. LoshinAmerican Ceramic Society . Lyle RubinAmerican Glaucoma Society Steve
20、n GeddeAmerican Optometric Association Karl CitekAmerican Society of Cataract and Refractive Surgery . Stephen KlyceContact Lens Institute. Stan RogaskiContact Lens Manufacturers Association . Martin DalsingDepartment of Veterans Affairs John TownsendFederated Cornea Societies. Michael BelinFood and
21、 Drug Administration CDRH/Division. Don CalogeroIndividual Ralph StoneNational Association of Optometrists and Opticians. Nick MiletiOpticians Association of America . Tom HicksOptical Laboratory Association. Steve SutherlinSunglass Association of America . Tibor GrossThe Vision Council Michael Vita
22、leUS ISO TC 172/SC7. Michael VitaleiiiThe Subcommittee on Instruments and Low Vision Devices, which modified thisAmerican National Standard, had the following members:William L. Brown, O.D., Ph.D., ChairMichael BelinDonald CalogeroCharles E. CampbellBruce DrumMatthew EverettThomas GarridoDavid Glass
23、erPriya JanakiramanTony KoStephen KlyceDavid LuceNick MiletiSharon MillerEli PeliMichael PflegerRobert RosenbergDexiu ShiJohn TownsendMichael VitaleThomas WhiteivAMERICAN NATIONAL STANDARD ANSI Z80.28-2017American National Standard for Ophthalmics Methods for Reporting Optical Aberrations of Eyes 1
24、1. Scope This standard specifies standardized methods for reporting the optical aberrations of eyes. 2. Normative References The following standards contain provisions which, through reference in this text, constitute provisions of this American National Standard. At the time of publication, the edi
25、tions indicated were valid. All standards are subject to revision, and parties to agreements based on this American National Standard are encouraged to investigate the possibility of applying the most recent editions of the standards indicated below. ISO 8429: 1986, Optics and Optical Instruments Op
26、hthalmology Graduated Dial Scale ANSI Z80.28-2017 2 3. Symbols and Definitions Symbols used in this standard are summarized in Table 1. Table 1 Symbols Symbol Name Section giving definition A(m,) Meridional term for magnitude/axis Zernike functions 5.1.9 mnc Zernike coefficient 3.3 cnm Zernike coeff
27、icient magnitude 5.1.9 m Meridional index for Zernike functions 3.2 )( mMmnMeridional term for Zernike functions 3.2.3 n Radial order index for Zernike functions 3.2 mnN Normalization term for Zernike functions 3.2.4 )(|mnR Radial term for Zernike functions 3.2.1 mnZ Zernike function (alternate nota
28、tion -Z(n,m) 3.2 Z Zernike function magnitude/axis form 5.1.9 Axis parameter for magnitude/axis form Zernike functions 5.1.9 Radial parameter for Zernike functions 3.2.2 Meridional parameter for Zernike functions 3.2.4 W Wavefront error in microns 3.4 (x,y) Measured gradient value at location x,y 3.
29、8 W(x,y) Wavefront gradient at a location x,y 3.8 fitGradient fitting error 5.3 3.1 Line of sight The line of sight is the line from the point of interest in object space to the center of the entrance pupil of the eye and continuing from the center of the exit pupil to the retinal point of fixation
30、(generally the foveola). 3.2 Zernike polynomial function A Zernike polynomial function is one of a complete set of functions defined and orthogonal over the unit circle; parameterized by a dimensionless radial parameter, , and a dimensionless meridional parameter ; designated by a non-negative radia
31、l integer index n and a signed meridional index m. Each Zernike polynomial function is the product of three terms a normalization term, a radial term and a meridional term so that a Zernike polynomial function with indices n and m is given by the equation: )()(| mMRNZmnmnmn The parameter is a real n
32、umber continuous over its range of 0 to 1.0. The parameter is a real number continuous over its range of 0 to 2. ANSI Z80.28-2017 3 Normalized Zernike polynomial functions are defined as orthogonal in the sense that they satisfy the following equation: ,1020, mmnnmnmndZZd where: n, n= 1 if n=n, n,n=
33、 0 if n does not equal n m,m= 1 if m=m, m.m= 0 if m does not equal m Un-normalized Zernike polynomial functions are defined as orthogonal in the sense that they satisfy the following equation: ,1020,0)1)(2(mmnnmnmnmdZZdn where: n, n = 1 if n=n, n,n= 0 if n does not equal n m,m = 1 if m=m, m.m = 0 if
34、 m does not equal m 0, m= 1 if m = 0, 0, m= 0 if m is not 0 3.2.1 Radial term The radial term for the Zernike polynomial function with indices n and m is given by the equation: |)|(5.002|)!|)|(5.0()!|)|(5.0(!)!()1()(mnssnsmnsmnsmnssnR where s is an integer summation index incremented by one unit. 3.
35、2.2 Radial parameter The radial parameter is a dimensionless number that takes values between 0 and 1. Its value at any radial distance, r, from the aperture center is given by the expression ar where a is the value of the aperture radius. 3.2.3 Meridional term The meridional term for a Zernike poly
36、nomial function with index m is given by the equations: M(m) = cos (m) if m 0 M(m) = sin (|m|) if m 0 NOTE: The meridional term is also known as the azimuthal term. ANSI Z80.28-2017 4 3.2.4 Meridional parameter - The meridional parameter is an angular value that takes values between 0 and 360 degree
37、s, expressed in the coordinate system defined in clause 4. NOTE: This is also called the azimuthal angle. 3.2.5 Normalization term If the Zernike polynomial function with indices n and m is defined as normalized, the normalization term is given by the equation: )1)(2(,0 nNmmn wher 0,m= 1 if m=0, 0,m
38、= 0 if m is not 0 If the Zernike polynomial function with indices n and m is defined as un-normalized, the normalization term is equal to1.0. 3.2.6 Normalized Zernike polynomial function A Zernike polynomial function is defined as normalized if the normalization term takes the form given in 3.2.5 fo
39、r “normalized” functions. 3.2.7 Un-normalized Zernike polynomial function A Zernike polynomial function is defined as un-normalized if the normalization term is equal to 1.0 3.2.8 Order The order of a Zernike polynomial function is the value of its radial index n. 3.3 Zernike coefficient A Zernike c
40、oefficient is a member of a set of a real numbers, mnc , that is multiplied by its associated Zernike function to yield a term that is subsequently used in a sum of terms to given a value equal to the best estimate of the surface, S(,), that has been fit with Zernike terms. Such a sum may be represe
41、nted by: mandnallmnmnZcS ),( 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. (See annex A for information on a method to find Zernike coefficients fro
42、m wavefront slope (gradient) data.) 3.3.1 Normalized Zernike coefficient A normalized Zernike coefficient is generated using normalized Zernike functions and is designed to be used with them to reconstruct a surface. Normalized Zernike coefficients have dimensional units of length. 3.3.2 Un-normaliz
43、ed Zernike coefficient An un-normalized Zernike coefficient is generated using un-normalized Zernike functions and is designed to be used with them to reconstruct a surface. Un-normalized Zernike coefficients have dimensional units of length. ANSI Z80.28-2017 5 3.4 Wavefront error W(x,y) or W(r,) Th
44、e wavefront error of an eye is the optical path-length (i.e., physical distance times refractive index) between a plane wavefront in the eyes entrance pupil and the wavefront of light exiting the eye from a point source on the retina. It is specified as a function (wavefront error function) of the (
45、x,y) or (r,) coordinates of the entrance pupil. 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. By convention, the wavefront error is set to zero at the pupil center by subtracting the central value from
46、 values at all other pupil locations. Wavefront error has physical units of meters (typically reported in micrometers) and pertains to a specified wavelength. 3.5 Optical path-length difference OPD The optical path-length difference at each point in a wavefront is the negative of the wavefront error
47、 (3.4). It represents the correction of the optical path length needed to correct the wavefront error. 3.6 Root mean square wavefront error (RMS error) The root mean square wavefront error for the eye is computed as the square root of the variance of the wavefront error function (3.4). Piston and av
48、erage tilt should be excluded from this calculation because they correspond to lateral displacements of the image rather than image degradation per se. RMS wavefront error is defined as: AdxdyyxWRMSpupilWFE2),(where A is the area of the pupil. If the wavefront error function is expressed in terms of
49、 normalized Zernike coefficients, the RMS error is equal to the square root of the sum of the squares of the coefficients with radial indices 2 or greater. mallnmnWFEcRMS,12)( NOTE: The RMS error may 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 may be found that this RMS value does not exactly m
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