Pupil Scaling for the Estimation of Aberrations in Natural Pupils

Pupil Scaling for the Estimation ofAberrations in Natural Pupils

Abbas Ommani*, Natalie Hutchings†, Damber Thapa‡, and Vasudevan Lakshminarayanan§

ABSTRACTPurpose. This study aimed to validate the mathematical Zernike pupil size scaling from bigger pupils to smaller pupils, andvice versa, by comparing the estimates of the Zernike coefficients with corresponding clinical measurements obtained atdifferent pupil sizes.Methods. The i.Profiler Plus (Carl Zeiss Vision, Inc, USA) was used to obtain measures of wavefront aberrations for twopupil sizes (3 mm and the maximum natural pupil size) from the right eyes of 28 visually normal subjects (mean [TSD] age,57 [T7] years) whose maximum pupil size was greater than or equal to 5 mm without pharmacological dilation. Zernikecoefficients were estimated for a 3-mm pupil size scaling down from the measured data of the maximum natural pupil sizeand, similarly, for the maximum pupil size scaling up from the measured data of the 3-mm pupil.Results. The differences between the estimated and measured values were not significantly different (repeated-measuresanalysis of variance; p 9 0.05) over the range of pupil sizes examined, irrespective of whether the estimates were made byscaling up from a small pupil or scaling down from a large pupil. However, the difference between the measured andestimated coefficients was more variable and less systematic when scaling to a larger pupil size when comparedwith scalingto a smaller pupil size.Conclusions. Estimation of ocular wavefront aberration coefficients either scaling down from large to smaller pupils orscaling up from smaller to large pupils provides estimates that are not significantly different from clinically measured values.However, when scaling up to a larger pupil size, the estimates are more variable. These findings have implications for pupilscaling on an individual basis, such as in cases of refractive surgery or when using pupil scaling to examine a clinical cohort.(Optom Vis Sci 2014;91:1175Y1182)

Key Words: Zernike coefficients, aberration, pupil size, refractive error, refractive surgery

The human eye presents with a continuum of aberrationsfrom simple spherical and cylindrical errors to complexhigher-order aberrations, such as spherical aberration, coma,

trefoil, and secondary astigmatism.1Y3 These all lead to retinalimage degradation.4 The magnitude of aberrations varies amongsubjects and eyes.1 Aberrations are the most prominent factor inreducing the quality of the retinal image when the pupil size ex-ceeds 3.0 mm in diameter1,3,4 and photopic visual acuity fallswith increasing pupil size.5 Several instruments, using a variety ofmethods, have been designed to measure the aberrations of thehuman eye over the past few decades and are widely available for

clinical applications. The Hartmann-Shack wavefront sensor6 isone such method and it measures the local slopes of various mea-surement points in the pupil allowing the original wavefront to bereconstructed from these slopes, usually using a least squares method.Deviation of the reconstructed wavefront from a reference planewavefront allows the wavefront aberrations to be quantified. Mostclinical aberrometers use Zernike polynomials to report the wave-front aberrations of the eye.7,8 The Zernike polynomials are a setof orthogonal basis functions over a circle of unit radius and aresuitable for accurately describing the wavefront aberration of anoptical system.9 Furthermore, International Organization for Stan-dardization standard ISO 24157 also recommended Zernike poly-nomials for reporting the optical aberrations of human eyes.10 TheZernike polynomials have the following three advantages. First, theyare mathematically independent (mutually orthogonal) from eachother. It is possible to perform statistical analysis or even removecertain terms from the data without affecting the contribution ofthe other terms. Second, each Zernike polynomial term representsa different type of aberration with its unique shape. The coefficient

1040-5488/14/9110-1175/0 VOL. 91, NO. 10, PP. 1175Y1182

OPTOMETRY AND VISION SCIENCE

Copyright * 2014 American Academy of Optometry

ORIGINAL ARTICLE

Optometry and Vision Science, Vol. 91, No. 10, October 2014

*MPhil†MCOptom, PhD‡MSc§PhD, FAAO

School of Optometry and Vision Science (AO, NH, DT, VL), Departments of

Physics, and Electrical and Computer Engineering (VL), University of Waterloo,

Waterloo, Ontario, Canada; and Department of Physics and Department of Electrical En-

gineering and Computer Science, University of Michigan, Ann Arbor, Michigan (VL).

Copyright © American Academy of Optometry. Unauthorized reproduction of this article is prohibited.

of each term indicates the amount of particular aberration presentin the eyes. Third, when Zernike polynomials are normalized, theabsolute value of the coefficient is the root-mean-square (RMS)contribution of that term, and that enables relative comparison ofZernike coefficients.11

One identified problem with using Zernike polynomials isthat they are pupil size dependent; therefore, the magnitude ofZernike coefficients changes when pupil size changes. This re-quires special consideration when measurements from differentinstruments or participants are compared7 to prevent ambiguousconclusions, particularly when the measurements are used forclinical applications such as laser refractive surgery.12 It is usuallymanaged by determining the Zernike coefficients for a commonpupil size. The algorithm used in an aberrometer uses a large num-ber of sensor elements for calculating the aberrations for the maxi-mum natural pupil and a smaller number of sensor elements for thesmaller pupil. An alternative approach that calculates the Zernikecoefficients from a fixed number of sensors and uses mathematicalpupil size scaling techniques to estimate the Zernike coefficientsat various pupil sizes has also been proposed.13Y21 Pupil scaling is amathematical procedure by which the measured Zernike coeffi-cients for a larger pupil size are estimated for a smaller pupil sizeand vice versa. Some studies22,23 have adapted this technique for pre-dicting wavefront aberration measurements at different pupil sizes.

Several recent studies have examined the relationship betweenrefractive error and Zernike coefficients of higher-order aberra-tions and have used pupil size scaling to scale the aberrations toa common pupil size, smaller than the natural pupil size.22,23

However, their research questions were not intended to examinethe pupil scaling technique itself. The scaling approach may pro-vide a more accurate estimation of wavefront aberration becauseit uses a larger set of sensor elements to extract Zernike coefficientsand it has been reported that data scaled for pupil size show greaterrepeatability of the Zernike coefficients than Zernike coefficientsderived only from the sensors contained within a given pupil size.12

The literature lacks the validation of pupil scaling techniques bycomparison with the measured clinical data. In this article, we val-idate the mathematical pupil size scaling formula by comparingthe estimates of the Zernike coefficients with corresponding clinicalmeasurements obtained at different pupil sizes.

METHODS

The study reviewed measurements taken with the i.Profiler Plusaberrometer (Carl Zeiss Vision, Inc, USA) in one of the battery oftests performed at the initial screening visit of a separate clinicalstudy, for which the pupil scaling procedure was not a researchquestion. The study sample for this experiment comprised righteyes of 28 visually normal presbyopic subjects (mean [TSD] age,57 [T7] years). The subjective spherical equivalents for right eyesof the participants ranged from j10.50 diopters (D) to +2.38 D(mean [TSD], j1.43 [T3] D). The maximum natural pupil sizesvaried among subjects and eyes and we included only those sub-jects with a pupil size greater than or equal to 5 mm. The meanmesopic natural pupil size of the sample was 5.39 mm and rangedbetween 5.00 and 6.26 mm. None of the participants had measure-ments taken with a pupil that had been dilated using a therapeuticpharmaceutical agent (i.e., a mydriatic or cycloplegic drop). Also,

none of the participants had undergone previous refractive surgeryor had any ocular pathology. The research followed the tenets of theDeclaration of Helsinki; informed consent was obtained from thesubjects after explanation of the nature and possible consequencesof the study, and the study, including all measurement procedures,was reviewed by and received ethics clearance through the Officeof Research Ethics at the University of Waterloo.

The i.Profiler Plus aberrometer, which is based on the Hartmann-Shack principle (LRef = 555 nm; maximum number of sensors,1500), was used to measure the wavefront aberrations in mesopicconditions up to the seventh order for the maximum natural pupildiameter and the device also provides corresponding data for a 3-mmpupil. It has the capability of obtaining measurements for pupilsizes of 2 to 7 mm. The measuring range of the wavefront sensor isfrom j20 DS to +20 DS for a 3.5-mm pupil size and j15 DS to +15 DS for a 5.5-mm pupil size, and up to 8 DC. In our study, themeasured data comprised Zernike coefficients obtained from thesensors contained within a 3-mm pupil diameter and those obtainedfrom the sensors contained within the maximum mesopic naturalpupil; both data sets were obtained directly from the subset of spotsin the Hartmann-Shack image and this was confirmed with thei.Profiler manufacturer (Meister D, Manager, Technical Market-ing Carl Zeiss Vision, Inc, personal communication [January 28,2014]). All measurements were obtained with a room illuminationof about 50 lux to allow for natural pupil dilation. During themeasurement process, the participants placed their chin on thechin rest and fixated on a target that served as an accommodativecontrol during the measurement period. The instrument auto-matically aligns the participant’s pupil center with the optical axisof the device before measurement acquisition. Measurements wereobtained for each eye separately for all participants and the testtypically took 30 seconds to complete for both eyes. However, weonly exploited measurements from the right eyes of the par-ticipants in this study.

In general, the wavefront aberration w(r,5) is represented asa linear combination of Zernike polynomials:

wðr; 5Þ ¼ ~n;m

Cmn Zm

n ðr=r max; 5Þ ð1Þ

where Zmn are the Zernike polynomials and Cm

n are their corre-sponding coefficients. The indices n and m represent the radialorder and azimuthal frequency of the azimuthal component, re-spectively. Similarly, r and 5 are the variables of the polar coordi-nate system and rmax is the maximum radial extent of the pupil.The rmax of different eyes or the same eye at different conditions aredifferent; hence, the measurements (i.e., Zernike coefficients) withdifferent pupil sizes cannot be directly compared.14 Let amn and bm

n

be the Zernike coefficients at pupil radius r1 and r2, respectively, forthe same wavefront aberration w(r,5) such that:

wðr; 5Þ ¼ ~n;m

amn Zmn ðr=r1; 5Þ ð2Þ

wðr; 5Þ ¼ ~n;m

bmn Zmn ðr=r2; 5Þ ð3Þ

The goal of the pupil scaling technique is to calculate Zernikecoefficients (bmn ) for a pupil diameter r2 from the measured Zernikecoefficients (amn ) for a pupil diameter r1. Schwiegerling14 solved this

1176 Estimation of Aberrations in Natural PupilsVOmmani et al.



problem by expressing bmn in terms of amn , r1, and r2 using algebraictechniques.

In this study, pupil scaling as described by Schwiegerling14 wasimplemented in MATLAB software version R2010b (MathWorks,Natick, USA) using the source code published by Ginis et al.12

Zernike coefficients were estimated for a 3-mm pupil size usingthe measured data from the maximum natural pupil size to providethe ‘‘scaled down estimate.’’ The corresponding Zernike coefficientestimates for the maximum pupil size, using the measured data fromthe 3-mm pupil size, provided the ‘‘scaled up estimate.’’ The esti-mated coefficients were then compared with the measured data.Only second-, third-, and fourth-order Zernike coefficients wereused for the analysis, as the coefficients were negligible for the fifthto seventh orders.

The data obtained directly from the aberrometer (measured)and the estimated data obtained from the pupil scaling techniquewere compared. The measured data were taken to be the ‘‘true’’value of the Zernike coefficients. Repeated-measures analysis ofvariance (RM-ANOVA) was used to evaluate overall differencesbetween estimated and measured Zernike coefficients (Contrasts:Method [Measured vs. Estimated] and Order [Lower vs. Higher]).For all analyses, > values less than or equal to 0.05 were consideredstatistically significant. Paired t tests were then used to evaluateindividual measured and estimated coefficients, with Holm cor-rection for multiple comparisons. Plots of the error versus themeasured coefficients were obtained for the second-, third-, andfourth-order Zernike coefficients, and the limits of agreement(1.96 � sample SD) were determined. The percentage of theabsolute error to the absolute mean for each coefficient was alsodetermined. All data analyses were carried out in the statisticalprogram R (v. 3.0.2).24

RESULTS

There was no significant difference between the overall esti-mated and measured Zernike coefficients for scaling up from the3-mm measured data to the maximum mesopic natural pupil sizeor for scaling down from the maximum mesopic natural pupilsize to 3 mm (RM-ANOVA; both p 9 0.5). There was a significantdifference in the lower- and higher-order coefficients (RM-ANOVA; both p G 0.001), but this was not different betweenthe estimated and measured methods of obtaining the coefficients(RM-ANOVA; Method*Order; both p 9 0.5). In the pairwisecomparison of each coefficient, there were no significant differ-ences for any coefficient (corrected paired t test; all p 9 0.05).

Pearson correlation coefficients between the estimated and mea-sured Zernike coefficients when scaling up from a 3-mm pupil sizeranged between 0.979 and 0.999, between 0.695 and 0.899, andbetween 0.734 and 0.898 for the second-, third-, and fourth-orderZernike coefficients, respectively. When scaling down from themaximum pupil size, Pearson correlation coefficients ranged be-tween 0.997 and 0.999, between 0.962 and 0.984, and between0.720 and 0.891 for the second-, third-, and fourth-order Zernikecoefficients, respectively.

Fig. 1 shows the error between the estimated and measuredsecond-order Zernike coefficients as a function of the magnitudeof the measured coefficients. Corresponding data for the third-and fourth-order coefficients are shown in Figs. 2 and 3. For all

coefficients, the mean error was centered at approximately zerofor both scaling up and scaling down by the pupil scaling proce-dure (Table 1). The widths of the limits of agreement and theabsolute percentage error were larger when the coefficients weredetermined by scaling up from measured data of a 3-mm to a largerpupil size than when scaling down from measured data of the largerpupil size to 3 mm (Table 1).

When scaling down, the data exhibited systematic smallerZernike coefficients irrespective of whether the coefficient was pos-itive or negative, illustrated by the long axis of the 95% confidenceellipse being tilted positively away from horizontal (Figs. 1 to 3). Nosuch clear, systematic error was observed when the estimates of theZernike coefficients were obtained by scaling up from the measured3-mm pupil data. There was no significant correlation between themagnitude of the difference between the measured and estimatedcoefficient and the pupil size, irrespective of whether the estimatedcoefficient was scaled up or down.

DISCUSSION AND CONCLUSION

This study describes a comparison of Zernike coefficients ob-tained directly from an aberrometer and that estimated using atheoretical pupil scaling formula. There was good to excellent cor-relation between the estimated and measured coefficients for allsecond-, third-, and fourth-order coefficients for both scaling upfrom a 3-mm to a larger pupil and scaling down from a larger pupilto 3 mm. The magnitude of the correlation generally reduced, inboth scaling cases, as the coefficient’s order increased. Although themean differences between the estimated and measured values foreach mode were not significantly different in both the scaling upand scaling down conditions, the differences were always largerwhen the Zernike coefficients were scaled up from a 3-mm to a largerpupil compared with the differences when scaling down to a 3-mmpupil. Therefore, taking the direct measures of the Zernike coef-ficients as the true estimate of the ocular aberrations, scaling downfrom a larger to a smaller pupil is more accurate than the converse.

Errors in pupil scaling from a smaller to a larger pupil may be aresult of several factors. Because the sensors of the Shack-Hartmandetector are in a regular, fixed spatial array, the number of sensorscontained within the true 3-mm pupil is far fewer than that con-tained within the true maximum pupil size. Fitting the Zernikepolynomials to the measured data from the 3-mm pupil will there-fore reduce the precision of, and thus introduce error in, the esti-mation of the coefficients both for 3 mm and when used to scale upto a larger pupil size. In addition, the pattern of higher-order co-efficients shows greater variation at the margin of the scaled up pupilsize and there may not be sufficient information in the data fromthe central 3-mm pupil to model this with precision. Our resultsindicate that this is the case; the scaled up data exhibit considerablyless agreement with the measured values (Table 1). Indeed, Giniset al.12 found that repeated estimation of the Zernike coefficientsfor higher-order aberrations directly from the sensors within the3-mm pupil had relatively low signal-to-noise ratio, whereas scal-ing down from a larger pupil to a 3-mm pupil did not. The noisemay be minimized if a dynamic aberrometer is used instead of sta-tic to measure the aberrations and the coefficients are extracted byaveraging multiple frames. Pupil scaling methods that deal with thescaled pupil being translated,15 rotated, and noncircular19,20 have been

Estimation of Aberrations in Natural PupilsVOmmani et al. 1177



developed, which can also contribute to inaccuracies in the estima-tion of Zernike coefficients; the methodology of this study assumedthat there was no translation of the pupil center and was unlikely tohave contributed to the errors in estimation seen.

With the availability of aberrometers in a clinical setting, thefindings of this study have clinical application where clinicians

need an estimation of Zernike polynomials for a fixed pupil sizebut when the available aberrometer measures only a single, dif-ferent pupil size. A previous, large study examined scaling up thepupil size from 5 mm in 0.25-mm increments, intending to ex-amine the implications for error in the estimation of wavefrontaberrations in the refractive surgery transition zone.25 The study

FIGURE 1.The error between the measured and estimated second-order Zernike coefficients Cj2

2 (top), C22 (middle), and C0

2 (bottom) for the pupil scaling procedureswhere data were scaled up from themeasured 3-mm data to themaximum pupil diameter (left panel) and scaled down from themaximumpupil size data toa 3-mm pupil diameter (right panel). The dotted ellipse illustrates a robust 95% confidence interval for the data. Open symbols represent a maximum pupilsize of less than 5.40 mm and filled symbols represent pupil sizes greater than 5.40 mm. The red dotted ellipse illustrates a robust 95% confidence intervalfor the data. Note the different scales for the ordinate axes of each panel and the abscissa for the bottom graphs.




found that when making estimations from 5 mm measured datafor pupil sizes 5.75 mm and larger, the level of RMS error wasgreater than the variability between repeated examinations. Sim-ilarly, Neal et al.26 found a good agreement between the rescaledand measured data over a small scaling range. They also found thatif the Zernike polynomials are poor representation of the data,there will be large RMS fit error and hence large pupil scaling

error.26 Our findings, although scaled up greater than or equal to2.0 mm from the starting pupil size, show a similar trend.

Bara et al.27 compared two pupil resizing methods: rescaling theZernike coefficients using the measured data from the originalpupil and refitting the wavefront slope measurements within thenew pupil based on the measurement provided by the aberro-meter. The results showed that rescaling generally provides better

FIGURE 2.The error between the measured and estimated third-order Zernike coefficients Cj3

3 (top), Cj13 (upper middle), C1

3 (lower middle), and C33 (bottom) for the

pupil scaling procedures where data were scaled up from the measured 3-mm data to the maximum pupil diameter (left panel) and scaled down from themaximum pupil size data to a 3-mm pupil diameter (right panel). Other details are as for Fig. 1.




results when scaling down to the smaller pupil whereas the re-fitting works well when scaling up to the larger pupil.27 The higherlimit of agreements while scaling up the Zernike coefficients in our

study is in agreement with Bara et al.27 A second implicationof this study is in the retrospective analysis of large databases.Several studies that examine the relationship between wavefront

FIGURE 3.The error between the measured and estimated fourth-order Zernike coefficients Cj4

4 (top), Cj24 (second row), C0

4 (third row), C24 (fourth row), and C4

4

(bottom) for the pupil scaling procedures where data were scaled up from the measured 3-mm data to the maximum pupil diameter (left panel) and scaleddown from the maximum pupil size data to a 3-mm pupil diameter (right panel). Other details are as for Fig. 1.




aberrations and refractive error have been published.22,23 To un-dertake meaningful between-subject comparisons, it is necessary toscale the data for each subject to an identical pupil size and for thepupil size to be large enough for the analysis to be generalizable.Typically, this is achieved by excluding any data sets with pupildiameters smaller than an arbitrary size. This might exclude a largepercentage of a database; for example,22 40% of a database was ex-cluded when the pupil diameter was set to 3.5 mm or larger. Itwould seem advantageous then to be able to scale up from smallerpupil sizes. However, our study demonstrates that although the scal-ing up procedure provides estimates that are not significantly differentfrom their corresponding measured values, the estimates will havea greater amount of variability and the error will not exhibit a sys-tematic error. Another approach to making between-subject com-parisons is to ensure all measurements are obtained with a large pupilsize by using a pharmacological agent to dilate the pupil. However, ithas been shown that the average RMS of the higher-order aberra-tions is different in eyes using different cycloplegic drops.28

In summary, the fact that Zernike coefficients are pupil size de-pendent impedes the direct comparison of coefficients measured bydifferent instruments and laboratories or from different partici-pants. The pupil scaling method can calculate the aberrations fora smaller pupil size based on the information from the maximumnatural pupil size (scaling down), or vice versa (scaling up). The scal-ing down method provides a better estimate of the refractive stateof the eye because it is not as impacted by large variations in theestimates, although we found no significant difference between themeasured and estimated coefficients over the range of pupils ex-amined, irrespective of whether the estimates were made by scalingup from a small pupil or the converse.

ACKNOWLEDGMENTS

The authors acknowledge Hoya Vision, Canada, which sponsored the clinicalexperiments for which these data were obtained as part of the baseline batteryof measurements.

These data were presented as a poster at the ARVO (Association for Re-search in Vision and Ophthalmology) meeting held on May 6, 2012, in FortLauderdale, FL, entitled ‘‘Validating Theoretical Pupil Size Scaling Formula

for the Estimation of Ocular Wavefront Aberrations’’ under the name of thesame authors of this article.

Received February 20, 2014; accepted June 16, 2014.

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TABLE 1.

Mean error and limits of agreement for the difference between the estimated and measured coefficients for each pupilscaling procedure

Scale up from 3 mm pupil Scale down from max pupil

Mean absolute error Mean absolute error

Order CoefficientMeanerror

Limits ofagreement

Absolute mean%

Meanerror

Limits ofagreement

Absolute mean%

Second Cj22 j0.003 T0.169 25.7 j0.00035 T0.024 9.8

C02 0.021 T0.297 4.92 0.00308 T0.041 7.8

C22 j0.022 T0.236 18.9 j0.00356 T0.031 2.3

Third Cj33 j0.030 T0.184 55.1 j0.00471 T0.023 23.3

Cj13 0.063 T0.220 75.8 0.00558 T0.020 31.3

C13 j0.020 T0.167 51.9 j0.00174 T0.015 21.8

C33 0.015 T0.213 84.0 0.00206 T0.020 38.6

Fourth Cj44 0.011 T0.089 74.7 j0.00134 T0.008 49.2

Cj24 j0.002 T0.092 115.8 0.00008 T0.009 65.0

C04 0.012 T0.160 50.9 j0.00098 T0.015 48.6

C24 j0.013 T0.126 117.2 0.00124 T0.012 69.2

C44 0.0117 T0.104 72.9 j0.00131 T0.011 57.2




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http://scien.stanford.edu/pages/labsite/2003/psych221/projects/03/pmaeda/index.html

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Abbas OmmaniSchool of Optometry and Vision Science

University of Waterloo200 University Ave W

Waterloo, ON, N2L 3G1Canada

e-mail: [email protected]




http://www.r-project.org/

Pupil Scaling for the Estimation of Aberrations in Natural Pupils

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