Zernike vs. Zonal Matrix Iterative Wavefront Reconstructor. Sophia I. Panagopoulou, PhD. University of Crete Medical School Dept.

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1 Zernie vs. Zonal Matrix terative Wavefront Reconstructor opia. Panagopoulou PD University of Crete Medical cool Dept. of Optalmology Daniel R. Neal PD Wavefront ciences nc. 480 Central.E. Albuquerque NM 873 Abstract Purpose: Matematical and clinical description of Zonal and Modal avefront reconstruction. Advantages and disadvantages of eac metod. Conclusions: Bot Modal and Zonal reconstruction can be an essential tool for evaluation of vision. Modal reconstruction generates smoot avefront maps and provides useful metrics. Zonal Reconstruction uses te entire data set and creates very detailed maps but doesn t provide quantitative interpretation or intuitive visualization. ntroduction Altoug ocular avefront aberrations ave been measured since 96 only recently it improved aberrometers did it become possible to routinely obtain avefront maps under clinical conditions. n refractive surgery apart from te long-establised corneal topograpy te Wavefront Analyzers are used for mapping te total aberrations of an eye troug te pupil entrance. At tis time te preferred surface fitting metod for caracterizing te avefront aberrations is te Zernie approximation ic uses te Zernie polynomials to describe and quantify te various avefront surfaces. Te conventional refraction can be decomposed into prism defocus cylinder and axis. n te same manner te remaining optical errors can be described it individual aberrations so-called Zernie modes it te process of Zernie decomposition. Te evaluation of individual Zernie modes reveals large differences in teir subective impact. 3 tudies from Applegate et al 3 and Williams et al 4 soed tat tis decomposition of te ave aberration in its fundamental components in order to approac te subective image quality could cause misinterpretations. Tis occurs because Zernie modes can interact strongly it eac oter and not in a - -

2 simple ay to determine te final image quality. Applegate et al 5 ave measured te interactions beteen different Zernie coefficients and found tat pairs of aberrations can sometimes increase te acuity from eac individual component as ell as lead to a larger decrease. Altoug te Zernie approximation is effective for describing te individual loer and iger order aberrations tey ave also a smooting effect tat can significantly limit te aberrations tat account for te avefront error. Due to limitations it Zernie (modal) reconstruction a different reconstruction metod suc as zonal reconstruction sould be considered. Here e ill present te concept for modal and zonal reconstructions. Wavefront reconstruction metods Te avefront at any point () is related to te gradients troug te vector gradient equation = i = i x y () ere i and are te unit vectors in te x and y directions. outel (980) 6 describes several metods for determining a avefront surface from slope data. He describes several different geometries in common use for obtaining slope data and introduces te concept of Modal and Zonal reconstructors. n te end te obective of eiter metod is to tae a set of avefront gradient measurements and and calculate or reconstruct te avefront map from tis data. Modal reconstructor n te modal reconstructor metod te avefront surface is described in terms of a set of smootly varying modes. Tese may be polynomials or oter functions or tey may be derived from some oter system properties (for example as used in adaptive optics). Te ey property of tese modes is tat an analytic derivative can be obtained tat may be fit to te measured slope data. Tus te avefront surface may be ritten as W ( x y) = CnPn ( x y). () n - -

3 ince te functions P n (xy) are continuous and analytic troug te first derivative te avefront gradient may be ritten ( x y) ( x x y) = y n n C C n n Pn ( x y) x Pn ( x y) y (3) Tis provides an analytic description of te avefront slope at every point (xy) ic may be compared to te measured gradient values to obtain te appropriate set of coefficients C n by least squares fitting or oter metods. Given a set of gradient data β x and β y at measurement point (x y ): A least squares fit to data may be obtained by minimizing te error function M M x Pn y Pn χ = β Cn β Cn = (4) n x n= y it respect to te coefficients C n. Once tese coefficients are determined te avefront surface is non troug Eq.. Eq. 4 may also be used to calculate te residual fit error ic is useful for caracterizing te quality of te fit or te appropriateness of te particular set of polynomials for describing te measured data. Tis type of reconstructor provides a very compact notation for describing te avefront since te detailed sape information is contained in te modes P n (xy). f te modes are cosen to be polynomials tat approximate pysical properties ten te coefficients may be interpreted to obtain a direct description of te pysical system. For example Zernie Polynomials are often used in optics because tey are closely related to te aberrations introduced in optical systems. Hence individual polynomials describe parameters suc as defocus astigmatism coma and sperical aberration. f ortogonal polynomials are used it is possible to separate te various effects since te presence (or absence) of one term does not affect te oters

4 Zonal reconstructor Te term zonal reconstructor comes about because rater tan describing te avefront in terms of a set of overall modes eac of ic is described analytically over te ole surface te avefront is described only locally over a limited zone. Tese zones are usually cosen to matc te area described by eac gradient measurement. Tey can represent a regular grid of square or rectangular elements or some oter local region suc as a exagonal or oter sape. For ac-hartmann avefront sensors tese zones are generally cosen to matc te lenslets altoug tis is not strictly necessary. Zonal reconstruction metods Tere are several metods for solving for a avefront surface tat point-by-point represents a selfconsistent solution to te gradient equation (Eq. ). Tese include te matrix iterative approac direct least-square fitting for avefront values and spline integration. Te body of tis paper ill concentrate on systems ere te slope data as obtained on a regular set of square regions and te average slope across eac region is non. Tis is te typical arrangement for ac-hartman avefront sensor data. Te same metods can be used for oter geometries but te appropriate nomenclature ould need to be developed. nterpolating functions For ac-hartmann gradient data te average slope across eac lenslet is determined from focal spot position sift. ince te slope may be different for an adacent lenslet it is reasonable to assume an interpolating function ere te avefront slope varies linearly beteen lenslets. Tus ( x0) = c c x and ( 0 y) = c c y. (5) 3 4 For a st order system e ill consider only tose adacent points tat are eiter directly above or belo or rigt and left of a central point. ntegrating (x0) and (0y) along te x and y axis results in te interpolating function for te avefront: ( x0) x = c0 cx c and ( 0 y) y = c0 c3 y c4. (6) - 4 -

5 Difference equations For a lenslet l it center (x l y l ) te average slope over te lenslet is ( l l ). t is more convenient to represent te data on a grid it indices () and spacing x and y. Tus te slope data is non at points and. Evaluating Eq. 5 for adacent points (for example at points () and ()) allos te determination of te coefficients: c c = = x c = 3 and c4 = y (7) =. (8) c 0 c0 = Tese coefficients can be determined for eac direction from te central point () i.e. () (-) () (-). Tis yields te folloing set of difference equations tat describe te avefront at a central point in terms of adacent points and te measured gradient values. y = ( ) = H { } (9) y = ( ) = H { } (0) { } = () x ( ) = H { } = () x ( ) = H H ere { } te values at. is a predictor operator tat predicts te avefront of from te measured slope and Boundary conditions Te boundary conditions must properly account for edge corner or interior points in te measurement. Generally for a ac-hartmann or oter slope sensor tere is also an independent measurement of te total amount of ligt tat as been received by te lenslet. For a fully illuminated lenslet tis corresponds - 5 -

6 - 6 - to te total of all te pixels (above some tresold corresponding to te camera bacground) covered by te lenslet. For a non-rectangular measurement space tere can be an aperture tat partially occludes some of te lenslet areas. Even for a circular pupil tere are a large number of edge lenslets tat ave a variable number of adacent elements. Tus e define a eigting function tat defines tese boundaries. Tis function as zero value everyere tat tere is no incident ligt. t may ave a value () in regions ere te lenslet is fully illuminated or some oter intermediate value for partially illuminated lenslets. Matrix iterative solution Te eigted average of te predictions from te four different directions is given by { } { } { } { } = H H H H. (3) Tis allos te value of te avefront to be predicted from avefront values in te four different directions. Gatering terms allos te definition of: ( ) [ ] ( ) [ ] = y y x x q (4) [ ] [ ] y x b = (5) g = (6) and g =. (7) Tis equation allos for iterative solutions for te avefront as given by: ( ) m m g q b ] [ ] [ = (8)

7 ere m is te iteration number. Equation 5 contains only tose points needed for an interior (uniform eigted) calculation ile te boundary conditions are contained in equation 4. An important consideration is te effect of edges on te data. For an edge or corner point te avefront can be predicted from only one or to directions. Hoever if te data is placed into a larger array tat as been initialized to zero ten te intensity values at boundary points are alays zero and te boundary conditions ill naturally be considered properly. t sould be noted tat for uniform intensity ere = everyere except at te boundaries tese equations converge to exactly te same equations as derived by outell. 6 t generally taes a number of iterations equal to te number of lenslets across te entire array to converge to a stable solution. For laser beams or oter avefronts ere te irradiance distribution varies rapidly e ave found tat it is possible to replace te eigting function it te measured irradiance over eac lenslet. Tis eigting serves to minimize te effect of increased noise as te irradiance decreases. Examples n Figure e ave analyzed te same data from te measured avefront of a uman eye using bot te modal and zonal metods. Te modal (a) (b) metod (a) produces very smoot surfaces Figure Comparison of Modal and Zonal metods for avefront reconstruction. (a) is modal reconstruction ile (b) is te zonal ile muc greater detail is evident in (b) te zonal metod. Te overall avefront P-V as approximately te same in bot cases. Wile te additional structure could be construed as noise it requires a priori information to mae tis determination. For tis case te structure is a combination of tear film and fixed structure in te ocular avefront measurement. n bot cases te sensor noise as lo - 7 -

8 enoug not to contribute significantly to te avefront image. Tus te structure represents a real avefront structure present in te measurement. n Figure (a) en e loo at te 0 t order Zernie avefront map tere is a smooting effect in te information but in (c) e can see a more detail map. (a) (b) (c) Figure Presentation of te iger order avefront map of an eye mont after conductive eratoplasty. a sos te Zernie 6 t order aberration map and c is te zonal avefront map from te same data. b sos te slit lamp poto of tis eye. Discussion Zernie modes can interact it eac oter to determine final image quality and te complexity of te interactions beteen modes means tat Zernie decomposition is not really effective as a metric of subective image quality. Modal representation as te advantage in tat it can provide quantization of clinical data. n modal reconstruction te feer te number of Zernie orders tat are used in a fit te smooter te representation ill be. On te oter and en te data is over-smooted te information is mostly lost especially for igly aberrated eyes

9 Te advantage of te zonal representation is in te lac of smooting effect due to te use of te entire ra data set resulting in representations it iger spatial frequencies. Tus zonal reconstruction may provide a better evaluation metod for calculating customized ablation profiles in eyes dominated by iger order aberrations as ell as for diagnostic applications. Combined information from bot metods could provide advantages in clinical evaluation of vision quality. Figure 3- Eye it corneal scars and te a and b represents te Zernie avefront map of 4 t and 0 t order analysis respectively. d sos te zonal avefront map tat corresponds it te corneal topograpy of tis eye in c. Generally zonal reconstructions lead to a unique description of te avefront surface at every measurement point. Te zonal reconstruction supports rapid (point by point) variation in te avefront surface and tus ill provide for very igresolution avefront description. t does not provide for a direct interpretation in optical terms. To determine defocus astigmatism or oter parameters from zonal data requires an additional computation step. n teory it is possible to fit very large basis sets it te modal suc tat te modal description can be used to describe very rapid local variations. Hoever for most solution metods tis ould require an inordinate amount of computer computation time. Te distinction beteen te to metods is useful to maintain and te solution metods are generally quite different. n practice bot metods are quite useful and it modern computers bot zonal and loer order modal may be calculated rapidly. Te difference - 9 -

10 beteen te avefronts derived from te to metods may provide useful insigt or interpretation of te information. References. mirnov M. Measurement of te ave aberration of te uman eye. Biofizia 96; 6:766:795.. Applegate RA arver EJ Kemsara V.Are all aberrations equal? J Refract urg. 00; Applegate RA Ballentine C Gross H arver EJ arver CA. Visual acuity as a function of Zernie mode and level of rms error. Optom Vis ci. 003; 80: Williams DR. Wat adaptive optics can do for te eye. Revie of Refractive urgery. 00; 3(3): mole MK Klyce D. Zernie polynomials are inadequate to represent iger order aberrations in te eye. nvest Optalmol Vis ci 003; 44: W. H. outell Wavefront estimation from ave-front slope measurements JOA 70(8) pp (980)