visual acuity as a function of zernike mode and level of...

ORIGINAL ARTICLE

Visual Acuity as a Function of Zernike Modeand Level of Root Mean Square Error

RAYMOND A. APPLEGATE, OD, PhD, FAAO, CHARLES BALLENTINE, BA, HILLERY GROSS, BA,EDWIN J. SARVER, PhD, and CHARLENE A. SARVER, BS

College of Optometry, University of Houston, Houston, Texas (RAA, HG), Department of Ophthalmology, The University of Texas HealthScience Center at San Antonio, San Antonio, Texas (CB), Sarver and Associates, Celebration, Florida (EJS, CAS)

ABSTRACT: Background. The coefficients of normalized Zernike expansion are orthogonal and reflect the relativecontribution of each mode to the total root mean square (RMS) wavefront error. The relationship between the level ofRMS wavefront error within a mode and its effect on visual performance is unknown. Purpose. To determine for variouslevels of RMS wavefront error how each mode of the normalized Zernike expansion for the second, third, and fourthorders affect high- and low-contrast acuity. Methods. Three healthy optimally corrected cyclopleged subjects readaberrated and unaberrated high- and low-contrast logarithm of the minimum angle of resolution acuity chartsmonocularly through a 3-mm artificial pupil. Acuity was defined by the total number of letters read correctly up to thefifth miss. Aberrated and unaberrated charts were generated using a program called CTView. Six levels of RMSwavefront error were used (0.00, 0.05, 0.10, 0.15, 0.20, and 0.25 �m). Each level of RMS error was loaded into eachmode of the second, third, and fourth radial orders individually for a total of 72 charts. Data were normalized bysubject, and the normalized data were averaged across subjects. Results. Across modes and within each mode as thelevel of RMS wavefront error increased above 0.05 �m of RMS wavefront error, visual acuity decreased in a linearfashion. Slopes of the linear fits varied depending on the mode. Modes near the center of the Zernike pyramid hadsteeper slopes than those near the edge. Conclusions. Increasing the RMS error within any single mode of thenormalized Zernike expansion decreases visual acuity in a linear fashion. The slope of the best fitting linear equationvaries with Zernike mode. Slopes near the center of the Zernike pyramid are steeper than those near the edge. Althoughthe normalized Zernike expansion parcels RMS error orthogonally, the resulting effects on visual performance asmeasured by visual acuity are not orthogonal. New metrics of the combined effects of the optical and the neuraltransfer functions that are predictive of visual performance need to be developed. (Optom Vis Sci 2003;80:97–105)

Key Words: ocular aberrations, visual optics, acuity, wavefront error, visual performance, Zernike expansion

With the introduction of fast and reliable wavefront sen-sors and high-speed low-cost computing, the correc-tion of the higher-order optical aberration of the eye is

being aggressively explored in the form of corneal refractive sur-gery, contact lenses, and intraocular lenses. Optical aberrations ofthe eye are typically described in terms of wavefront error. Wave-front error is the difference between the ideal wavefront and theactual wavefront error of the optical system as a function of loca-tion within the exit pupil.

An Optical Society of America task force has recommended thenormalized Zernike expansion1 as the standard method for speci-fying wavefront error of the eye (Fig. 1). One advantage of thenormalized Zernike expansion is that the absolute value of eachmode coefficient represents the root mean square (RMS) wave-

front error attributable to that mode. Consequently, by simplyscanning the values of the coefficients, one can quickly identify themode or modes having the greatest impact on the total RMS wave-front error of the eye.

Although, the normalized coefficients reveal the relative contri-butions of each Zernike mode to the total wavefront error,1 they donot reveal the relative impact of each Zernike mode on visualacuity. For example, Applegate et al.2 demonstrated that for a fixedlevel of RMS error (0.25 �m over a 6-mm pupil), not all modes ofthe Zernike expansion induce equivalent losses of visual functionas measured by high- and low-contrast acuity. In addition, a com-munication from David Williams and his group revealed that theyhave qualitatively found the same effects we reported2 using adeformable mirror and an experimental paradigm that utilized blur

1040-5488/03/8002-0097/0 VOL. 80, NO. 2, PP. 97–105OPTOMETRY AND VISION SCIENCECopyright © 2003 American Academy of Optometry

Optometry and Vision Science, Vol. 80, No. 2, February 2003

matching. Here we explore how the level of the RMS error withineach mode influences high- and low-contrast logarithm of theminimum angle of resolution (log MAR) visual acuity.

The goal of the present study was to determine for various levelsof RMS error how each mode of the normalized Zernike expansionfor the second, third, and fourth radial orders affect high- andlow-contrast acuity.

METHODS

We considered two experimental approaches. The first is a directmethod where a subject’s aberrations are first minimized using adeformable mirror (a deformable mirror typically reduces the eye’saberrations over a 6.3-mm pupil to about 0.1 �m [A. Roorda,personal communication]), and then the desired aberration isadded to the mirror and the subject is asked to read a traditional logMAR acuity chart. The second is to aberrate the log MAR acuitychart with the desired aberration and have a subject (whose ocularaberration has been minimized) read the aberrated chart followingmethods reported earlier.2, 3 The first method requires a well-cal-ibrated deformable mirror (expensive) and the latter a computerand a high-quality printer (inexpensive). We choose the latter ex-perimental approach as in our previous work.2, 3

Subjects

Three healthy volunteers who were free of significant ocular andsystemic pathology and were between the ages of 25 and 52 years ofage with 20/16 or better acuity served as subjects. Two of the threesubjects had their ocular aberrations measured using a laboratoryShack-Hartmann wavefront sensor. The third subject’s ocular ab-

erration was not measured. The coefficients for a normalizedZernike expansion through the 10th order for the two measuredsubjects are provided in Tables 1 and 2 along with their associatedphase transfer functions (see Figs. 2 and 3).

Generation of Aberrated Acuity Charts

As in our previous work,2, 3 a commercially available programcalled CTView was used to generate both high- and low-contrastaberrated and unaberrated log MAR acuity charts. CTView intro-duces the aberration into the charts by directly setting the wavefrontZernike coefficient values and performing a convolution of the result-ing point-spread function with an image of an acuity chart. Each chartwas created using a random selection of letters from an equally iden-tifiable letter set, scaled for a 10-ft test distance. Charts were printed on8.5- � 11-in sheets of photographic grade paper using a high-resolu-tion printer (600 dpi). High- and low- (11%) contrast charts weregenerated with between 0 and 0.25 �m of RMS wavefront error(0.00, 0.05, 0.10, 0.15, 0.20, and 0.25 �m). Each level of RMSwavefront error was individually loaded into each Zernike mode forthe second (�C2

�2, C20, C2

2�), third (�C3�3, C3

�1, C31, C3

3�), and fourthradial order (�C4

�4, C4�2, C4

0, C42, C4

4�) of the normalized Zernike ex-pansion. The acuity measured for the zero RMS wavefront error con-ditions (12) for each subject served as the normalizing reference forthat subject (see below). Thus, a total of 144 different charts (6 levelsof RMS error � 12 Zernike modes � 2 contrast levels) were used inthe study.

For clinical reference, an equivalent spherical defocus for a givenRMS wavefront error can be calculated using the following generalformula for calculating sphere (S), cylinder (C), and axis (�) from

FIGURE 1.Pictorial depiction of the second-, third-, and fourth-radial order Zernike modes.

98 Visual Acuity and Zernike Aberrations—Applegate et al.


the second radial order coefficients of the Zernike expansion fit towavefront error.

S � �4�3C2

0 � 2�6 ��C22�2 � �C2

�2�2

R2 (1)

C � �4�6 ��C2

2�2 � �C2�2�2

R2 (2)

� � tan�1�C2�2�C2

2��2 (3)

In Equations 1 to 4, coefficients Cnm are Zernike mode coeffi-

cients expressed in micrometers. R is the radius of the pupil ex-pressed in millimeters. S is the dioptric power of the sphere ex-pressed in diopters. C is the dioptric power of the cylinder, and �is the axis of the cylinder in degrees.

For the purposes of calculating equivalent defocus where thetotal RMS error is assumed to exist in the spherical defocus term (C2

0), Equation 2 reduces to

Equivalent defocus � �4�3C2

0

R2

Using the equivalent defocus formula (Equation 4), 0.00, 0.05,0.10, 0.15, 0.20, and 0.25 �m of RMS wavefront error over a6-mm pupil corresponds to an equivalent defocus of 0.000, 0.038,0.077, 0.115, 0.154, and 0.192 D, respectively. Chart appearancefor no wavefront error to 0.25 �m of RMS wavefront error loadedinto the defocus term (C2

0) is illustrated in Fig. 4.

Protocol

The optical quality of the normal eye is maximized when the eyeis optimally corrected using a traditional spherocylindrical correc-tion, and the pupil is limited to approximately 3 mm.4 Stateddifferently, a 3-mm artificial pupil optimizes the normal eye’s op-tical quality by balancing diffraction effects that result with smallerpupils and higher-order aberrations that are passed with largerpupils. To illustrate why it is important to optimize the optical

TABLE 1.Average normalized Zernike expansion coefficients for thehigher-order (third through tenth) aberrations over a 3-mmpupil for subject RAA from five Hartmann-Shack wavefrontmeasurementa

n m �m n m �m

3 �3 �5.78E�02 8 �8 4.93E�053 �1 2.13E�02 8 �6 �5.97E�053 1 3.82E�03 8 �4 3.20E�053 3 �2.04E�02 8 �2 1.75E�064 �4 3.39E�03 8 0 �1.27E�054 �2 4.67E�04 8 2 �1.80E�054 0 3.30E�03 8 4 4.17E�054 2 1.12E�03 8 6 6.70E�054 4 3.43E�03 8 8 �7.41E�055 �5 �9.91E�04 9 �9 �3.21E�065 �3 5.23E�03 9 �7 �1.76E�065 �1 �2.99E�03 9 �5 �1.09E�065 1 �4.63E�04 9 �3 4.28E�065 3 2.11E�03 9 �1 �1.03E�065 5 8.89E�04 9 1 �2.33E�076 �6 1.13E�03 9 3 1.27E�066 �4 �4.80E�04 9 5 2.82E�066 �2 1.13E�04 9 7 �4.87E�066 0 4.04E�04 9 9 2.65E�066 2 4.05E�04 10 10 �5.81E�076 4 �7.08E�04 10 �8 �6.92E�076 6 �1.33E�03 10 �6 1.09E�067 �7 1.04E�04 10 �4 �6.72E�077 �5 5.09E�05 10 �2 �1.44E�077 �3 �2.47E�04 10 0 1.79E�087 �1 1.02E�04 10 2 3.38E�077 1 1.12E�05 10 4 �7.93E�077 3 �9.02E�05 10 6 �1.08E�067 5 �1.11E�04 10 8 1.15E�067 7 2.92E�04 10 10 �1.58E�07

a Units are in �m; n is the radial order, and m is the angularfrequency of each Zernike mode. Total higher-order root meansquare error over a 3-mm pupil is 0.066 �m.

TABLE 2.Average normalized Zernike expansion coefficients for thehigher-order (third through 10th) aberrations over a 3-mmpupil for subject HG from five Hartmann-Shack wavefrontmeasurementa

n m �m n m �m

3 �3 �3.55E�02 8 �8 3.92E�063 �1 1.96E�03 8 �6 4.55E�053 1 1.17E�02 8 �4 5.82E�063 3 9.79E�03 8 �2 4.54E�054 �4 1.68E�03 8 0 �2.90E�054 �2 8.39E�03 8 2 �5.20E�064 0 7.50E�03 8 4 �7.26E�064 2 �2.59E�03 8 6 4.05E�054 4 �1.68E�03 8 8 �5.09E�055 �5 1.83E�03 9 �9 2.43E�065 �3 3.06E�03 9 �7 �1.26E�065 �1 1.20E�03 9 �5 5.86E�085 1 �3.96E�04 9 �3 2.85E�065 3 �1.15E�03 9 �1 �4.00E�085 5 �2.10E�03 9 1 �9.00E�076 �6 �9.24E�04 9 3 �1.95E�066 �4 �2.89E�04 9 5 �2.63E�076 �2 �8.91E�04 9 7 �1.27E�066 0 1.06E�03 9 9 4.68E�086 2 1.34E�06 10 10 1.21E�076 4 2.04E�04 10 �8 �5.48E�086 6 �7.11E�04 10 �6 �8.44E�077 �7 9.78E�05 10 �4 7.40E�087 �5 �5.65E�05 10 �2 �8.65E�077 �3 �1.64E�04 10 0 1.19E�077 �1 �2.28E�05 10 2 2.56E�077 1 3.48E�05 10 4 5.82E�087 3 7.98E�05 10 6 �7.56E�077 5 5.83E�05 10 8 1.05E�067 7 �2.18E�06 10 10 �7.26E�07

a Units are in �m; n is the radial order, and m is the angularfrequency of each Zernike mode. Total higher-order root meansquare error over a 3-mm pupil is 0.041 �m.

Visual Acuity and Zernike Aberrations—Applegate et al. 99


quality of the eye, Fig. 5 shows two-dimensional modulation trans-fer functions (MTF’s) for an eye with a 3-mm diffraction-limitedpupil (Fig. 5A); subject RAA’s eye with 0.066 �m of RMS over a3-mm pupil (Fig. 5B); and in Fig. 5 C to H, the two dimensionalmodulation transfer functions for a simulated 6-mm eye with 0.25

(Fig. 5C), 0.20 (Fig. 5D), 0.15 (Fig. 5E), 0.10 (Fig. 5F), 0.05 (Fig.5G), and 0.00 (Fig. 5H) �m of RMS wavefront error loaded intoC2

0(defocus). Important for this experiment is to notice in Fig. 5that 0.25 down to 0.15 �m of RMS decreases the MTF such thata real eye viewing through a 3 mm pupil can see all relevant spatial

FIGURE 2.Phase transfer function calculated from subject RAA’s higher-order aberrations (third through 10th radial orders) using ZMAX. A value of 200 lines/mmcorresponds to approximately 20/10 assuming a posterior nodal distance of 16.67 mm.

FIGURE 3.Phase transfer function calculated from subject HG’s higher-order aberrations (third through 10th radial orders) using ZMAX. A value of 200 lines/mmcorresponds to approximately 20/10 assuming a posterior nodal distance of 16.67 mm.



frequencies (Fig. 5B). For the 0.10 RMS error condition, the realeye’s 3-mm pupil passes approximately the same spatialinformation.

Although each individual Zernike mode affects the two-dimen-sional MTF differently, modes near the center of the Zernike treeadversely affect the MTF more than those near the edge. In thisstudy and in our prior work,2 defocus (C2

0), spherical aberration (C4

0), and secondary astigmatism (C4�2, C4

�2) have the largest impacton the MTF. The general rule illustrated in Fig. 5 for defocus holdstrue for all Zernike modes tested. That is, the vast majority of thespatial information contained in the charts are passed through the

3-mm pupil as long as the RMS error is �0.10 �m. Consequently,one can examine the impact of 0.25 down to 0.10 �m of aberra-tion over a 6-mm pupil with minimal to no loss in fidelity througha 3-mm pupil. For the 0.05 test condition, the real eye’s 3-mmcondition will not pass the vast majority of the relevant spatialinformation. Therefore, the results of the 0.05-�m RMS wave-front error test condition are anticipated to be nearly equivalent tothe 0.00-�m RMS normalizing condition. Said differently, the3-mm pupil is now the limiting factor for the 0.05-�m test con-dition and not the aberration placed in the chart.

To maximize the optical quality, the subject’s test eye was di-

FIGURE 4.Chart appearance for zero root mean square wavefront error to 0.25 �m of root mean square wavefront error in 0.05-�m steps loaded into the defocusterm (C2

0) over a 6 mm pupil. In terms of spherical dioptric defocus, these levels correspond to 0.000, 0.038, 0.077, 0.115, 0.154, and 0.192 D. Thesmall bars at the side of each chart denote 0.0 logMAR (20/20). Aberrated charts were generated using CTView (www.sarverassociates.com).

FIGURE 5.Two-dimensional modulation transfer function for a diffraction-limited simulated eye with a 3-mm pupil (A), higher-order aberration over a 3-mm pupilfor subject RAA’s eye (B), and a simulated eye with a 6-mm pupil having 0.25 �m (C) down to 0.00 �m (H) of root mean square error loaded into Zenikemode C2

0 (C–H) in 0.05 �m steps. A value of 200 cycles/mm is approximately equivalent to 20/10, and 100 cycles/mm is approximately equivalent to20/20 assuming a secondary nodal-to-retina distance of 16.67 mm. Color contour steps are in 0.1 increments. Two-dimensional modulation transferfunctions were created using CTView (Sarver and Associates, Celebration, FL, www.sarverassociates.com).



lated with 1% cyclopentolate, and the subject viewed each of thecharts through a 3-mm artificial pupil aligned to the eye’s achro-matic axis. Artificial pupil alignment was achieved using a fovealachromatic alignicator5–7 and maintained using a bite barmounted to a three-dimensional vise. The fellow eye was occluded.The test eye was refracted to maximum plus to best visual acuity forthe 10-ft test distance through the aligned 3-mm artificial pupil.Each subject read each of the charts in random order three timesuntil five letters were missed. The total number of letters read

correctly up to the point of the fifth miss was recorded for eachchart. Chart illumination was maintained at 100 cd/m2.

RESULTS

To compare data across subjects, the data for each subject wasfirst normalized to the mean of the letters read correctly whileviewing the unaberrated chart.

FIGURE 6.Mean number of letters lost for the 12 Zernike modes tested as a function of root mean square (RMS) wavefront error. Error bars are �1 SD.

FIGURE 7.Test condition with the best linear fit to letters lost as a function of root mean square (RMS) wavefront error (left) and test condition with the worst linearfit to letters lost as a function of RMS wavefront error (right). Error bars are �1 SD.



L (gained or lost) � LC(A) � LC(UA) (5)

where L letters gained or lost

LC�A� � letters read correctly on the aberrated chart

LC (UA) � average letters read correctly on the unaberrated chart

Using Equation 5, a negative L means that the subject lost letterscompared with the mean of the unaberrated condition. To illus-trate, say that the total number of letters read correctly up to thefifth miss under the unaberrated condition on the 36 trials aver-aged 64.67 letters read correctly (i.e., LC(UA) 64.67). Now, saythat for the C2

0(defocus) 0.25-�m test condition, the number ofletters read correctly for the three trials were 58, 56, and 55. UsingEquation 5 yields letters lost for the three trials of �6.67, �8.67,and �9.67 and a mean letters lost of �8.33. Variability in thenumber of letters read correctly for each test condition was smallfor all subjects. To illustrate the precision with which the subjectscould read the aberrated and unaberrated charts, the average stan-dard deviation in letters lost across all test conditions for the threesubjects were, 0.91 (RAA), 1.05 (HG), 1.18 (BB).

Consistent with our anticipation, the 0-�m RMS error condi-tions and the 0.05-�m RMS conditions were not significantlydifferent by paired t-test (p 0.05). All other RMS levels weresignificantly different from 0 RMS conditions and from each otherat p 0.01 (Fig. 6 and Table 3). The two most probable explana-tions for the finding that the 0-�m RMS and 0.05-�m RMSconditions were not significantly different are (1) the chart con-tained more spatial information than can pass through a 3-mmartificial pupil (as discussed above) and (2) very low levels of RMSerror make no significant difference in our ability to measure high-contrast visual acuity. We include the RMS 0.05-�m data in thelinear regressions below; it should be remembered that that thiscondition is essentially equivalent to the normalizing 0-�m RMSerror condition.

The mean number of letters gained or lost for each subject foreach test condition was averaged across subjects to determine themean of means for each test condition. For each Zernike mode, themean of means was, in turn, plotted as a function of the RMSwavefront error.

Fig. 7 illustrates the experimental Zernike mode yielding thebest (Fig. 7, left) and worst (Fig. 7, right) linear fit to the data.Intercepts, slopes, and r2 are displayed in a Zernike pyramid for thehigh- (Fig. 8) and low- (Fig. 9) contrast charts, respectively. Noticein Figs. 8 and 9 that Zernike modes near the center of the pyramidhave the greatest impact on acuity as a function of RMS error (thesteepest slopes).

DISCUSSION

The coefficients of the normalized Zernike expansion revealthe relative contributions of each Zernike mode to the totalRMS error. They do not reveal their relative contribution tovisual performance as measured by high- or low-contrast logMAR acuity. As RMS error increases by �0.05 �m, high- andlow-contrast acuity decreases and is well described by a linearfunction across Zernike modes (Fig. 6) or for any single Zernikemode (Figs. 7, 8, and 9).

The actual degree of independence of each mode in the formationof the retinal image is not revealed in this study because each mode wastested independently. In fact, in studies we allude to in the discussionof our earlier work2 and later reported (R. Applegate, paper presentedat [email protected], [email protected], @berrometry.online An-nual Meeting, Milan, Italy, September 2002. CD of the meeting to bedistributed by EyeWorld in 2003), we show acuity data demonstrat-ing that aberrations from different modes interact. As we stated in thepublished paper,2 “When combined, Zernike modes can interact toimprove acuity despite the increase in total wavefront error. For exam-ple, spherical aberration and defocus can be combined in such a man-ner that the individual modes affect vision more than the combina-tion. Likewise, astigmatism and secondary astigmatism can becombined such that their summed effect on acuity is less than theindividual modes. Modes two radial orders apart (e.g., radial orders 2and 4) and having the same angular frequency (e.g., 0, �2, or 2) canbe combined such that the combined effect on acuity is less than theindividual effects.” Furthermore, Zernike modes within the same ra-dial order can interact to decrease visual performance to a greaterextent than loading the same total RMS error into a single term (R.Applegate, reported at [email protected], [email protected], @ber-rometry.online Annual Meeting, Milan, Italy, September 2002. CDof the meeting to be distributed by EyeWorld in 2003; this work hasbeen accepted for publication).8

New metrics (besides RMS error) that better correlate withvisual performance need to be developed. To this end, LarryThibos’ laboratory, David Williams’ laboratory, and our labo-ratory have joined forces to explore a variety of optical metricsand combined optical neural metrics that better correlate withmeasured visual performance. Fortunately, this endeavor ismade easier because carefully measured wavefront error funda-mentally defines the optical properties of the system. A directconsequence is that wavefront error can be used to calculateother optical metrics in the pupil, spatial, or frequency do-

TABLE 3.Paired t-test comparing the differences in means for all sixlevels of root mean square wavefront error tested

Mean Difference DF t Value p Value

C 00, C 05a 0.556 107 1.741 0.0846C 00, C 10 1.500 107 4.820 0.0001C 00, C 15 2.741 107 9.046 0.0001C 00, C 20 4.315 107 11.924 0.0001C 00, C 25 5.704 107 16.337 0.0001C 05, C 10 0.944 107 3.348 0.0011C 05, C 15 2.185 107 7.661 0.0001C 05, C 20 3.759 107 11.244 0.0001C 05, C 25 5.148 107 15.618 0.0001C 10, C 15 1.241 107 4.699 0.0001C 10, C 20 2.815 107 8.808 0.0001C 10, C 25 4.204 107 15.034 0.0001C 15, C 20 1.574 107 5.120 0.0001C 15, C 25 2.963 107 9.622 0.0001C 20, C 25 1.389 107 4.671 0.0001

a C 00 the 0.00-�m wavefront error condition, C 05 the0.05-�m wavefront error condition, etc.



mains. Capitalizing on this fact, we are in the process of evalu-ating over 20 optical metrics calculated from the wavefronterror against the visual performance results reported here andelsewhere2 and will be reporting these results in the near future.The most promising appear to be metrics that include compen-sation for both the optical and neural transfer functions. Welook forward to completing these studies and reporting them inthe near future.

CONCLUSIONS

Increasing the RMS error within any single mode of the nor-malized Zernike expansion decreases visual acuity in a linear fash-ion. The slope of the best fitting linear equation varies withZernike mode. Slopes near the center of the Zernike pyramid aresteeper than those near the edge. Although the normalized Zernikeexpansion parcels RMS error orthogonally, the resulting effects on

FIGURE 8.Intercept, slope, and r2 values for each Zernike mode by location in the Zernike pyramid for high-contrast log MAR charts.

FIGURE 9.Intercept, slope, and r2 values for each Zernike mode by location in the Zernike pyramid for low-contrast log MAR charts.



visual performance as measured by visual acuity are not orthogo-nal. New metrics of the combined effects of the optical and theneural transfer functions that are predictive of visual performanceneed to be developed.

ACKNOWLEDGMENTS

This work was supported by National Eye Institute, National Institutes ofHealth grant R01 08520 to RAA, National Eye Institute, National Institutesof Health grant R44 EY 11681 to EJS, HEAF Funds awarded to RAA fromthe University of Houston, The Visual Optics Institute at the College of Op-tometry, University of Houston, and unrestricted funds awarded to the Uni-versity of Texas Health Science Center San Antonio, Department of Ophthal-mology from Research to Prevent Blindness. Results of this study were firstreported by C. Ballentine, et al. at the Association for Research in Vision andOphthalmology, Ft. Lauderdale, FL, May 2002.

R. A. Applegate, E. J. Sarver, and C. A. Sarver have proprietary interest inthe program CTView.

Received May 6, 2002; revision received November 22, 2002.

REFERENCES

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Vol 35. OSA Technical Digest Series. Washington, DC: Optical So-ciety of America, 2000:232–44.

2. Applegate RA, Sarver EJ, Khemsara V. Are all aberrations equal? JRefract Surg 2002;18:S556–62.

3. Doshi JB, Sarver EJ, Applegate RA. Schematic eye models for simula-tion of patient visual performance. J Refract Surg 2001;17:414–9.

4. Charman WN. Wavefront aberration of the eye: a review. Optom VisSci 1991;68:574–83.

5. Thibos LN, Bradley A, Still DL, Zhang X, Howarth PA. Theory andmeasurement of ocular chromatic aberration. Vision Res 1990;30:33–49.

6. Simonet P, Campbell MC. The optical transverse chromatic aberra-tion on the fovea of the human eye. Vision Res 1990;30:187–206.

7. Rynders M, Lidkea B, Chisholm W, Thibos LN. Statistical distribu-tion of foveal transverse chromatic aberration, pupil centration, andangle psi in a population of young adult eyes. J Opt Soc Am (A)1995;12:2348–57.

8. Applegate RA, Marsack J, Ramos R. Interaction between aberrationscan improve or reduce visual performance. J Cataract Refract Surg, inpress.

Raymond A. ApplegateCollege of Optometry

University of HoustonHouston, TX 77204-2020e-mail: [email protected]



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