1040-5488/03/8002-0097/0 VOL. 80, NO. 2, PP. 97–105
OPTOMETRY AND VISION SCIENCE
Copyright © 2003 American Academy of Optometry
ORIGINAL ARTICLE
Visual Acuity as a Function of Zernike Mode
and 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 Health
Science 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 relative
contribution of each mode to the total root mean square (RMS) wavefront error. The relationship between the level of
RMS wavefront error within a mode and its effect on visual performance is unknown. Purpose. To determine for various
levels of RMS wavefront error how each mode of the normalized Zernike expansion for the second, third, and fourth
orders affect high- and low-contrast acuity. Methods. Three healthy optimally corrected cyclopleged subjects read
aberrated and unaberrated high- and low-contrast logarithm of the minimum angle of resolution acuity charts
monocularly through a 3-mm artificial pupil. Acuity was defined by the total number of letters read correctly up to the
fifth miss. Aberrated and unaberrated charts were generated using a program called CTView. Six levels of RMS
wavefront 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 each
mode of the second, third, and fourth radial orders individually for a total of 72 charts. Data were normalized by
subject, and the normalized data were averaged across subjects. Results. Across modes and within each mode as the
level of RMS wavefront error increased above 0.05 ␮m of RMS wavefront error, visual acuity decreased in a linear
fashion. Slopes of the linear fits varied depending on the mode. Modes near the center of the Zernike pyramid had
steeper slopes than those near the edge. Conclusions. Increasing the RMS error within any single mode of the
normalized Zernike expansion decreases visual acuity in a linear fashion. The slope of the best fitting linear equation
varies with Zernike mode. Slopes near the center of the Zernike pyramid are steeper than those near the edge. Although
the normalized Zernike expansion parcels RMS error orthogonally, the resulting effects on visual performance as
measured by visual acuity are not orthogonal. New metrics of the combined effects of the optical and the neural
transfer 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
W
ith the introduction of fast and reliable wavefront sensors and high-speed low-cost computing, the correction of the higher-order optical aberration of the eye is
being aggressively explored in the form of corneal refractive surgery, contact lenses, and intraocular lenses. Optical aberrations of
the eye are typically described in terms of wavefront error. Wavefront error is the difference between the ideal wavefront and the
actual wavefront error of the optical system as a function of location within the exit pupil.
An Optical Society of America task force has recommended the
normalized Zernike expansion1 as the standard method for specifying wavefront error of the eye (Fig. 1). One advantage of the
normalized Zernike expansion is that the absolute value of each
mode coefficient represents the root mean square (RMS) wave-
front error attributable to that mode. Consequently, by simply
scanning the values of the coefficients, one can quickly identify the
mode or modes having the greatest impact on the total RMS wavefront error of the eye.
Although, the normalized coefficients reveal the relative contributions of each Zernike mode to the total wavefront error,1 they do
not reveal the relative impact of each Zernike mode on visual
acuity. For example, Applegate et al.2 demonstrated that for a fixed
level of RMS error (0.25 ␮m over a 6-mm pupil), not all modes of
the Zernike expansion induce equivalent losses of visual function
as measured by high- and low-contrast acuity. In addition, a communication from David Williams and his group revealed that they
have qualitatively found the same effects we reported2 using a
deformable mirror and an experimental paradigm that utilized blur
Optometry and Vision Science, Vol. 80, No. 2, February 2003
98
Visual Acuity and Zernike Aberrations—Applegate et al.
FIGURE 1.
Pictorial depiction of the second-, third-, and fourth-radial order Zernike modes.
matching. Here we explore how the level of the RMS error within
each mode influences high- and low-contrast logarithm of the
minimum angle of resolution (log MAR) visual acuity.
The goal of the present study was to determine for various levels
of RMS error how each mode of the normalized Zernike expansion
for the second, third, and fourth radial orders affect high- and
low-contrast acuity.
METHODS
We considered two experimental approaches. The first is a direct
method where a subject’s aberrations are first minimized using a
deformable mirror (a deformable mirror typically reduces the eye’s
aberrations over a 6.3-mm pupil to about 0.1 ␮m [A. Roorda,
personal communication]), and then the desired aberration is
added to the mirror and the subject is asked to read a traditional log
MAR acuity chart. The second is to aberrate the log MAR acuity
chart with the desired aberration and have a subject (whose ocular
aberration has been minimized) read the aberrated chart following
methods reported earlier.2, 3 The first method requires a well-calibrated deformable mirror (expensive) and the latter a computer
and a high-quality printer (inexpensive). We choose the latter experimental approach as in our previous work.2, 3
Subjects
Three healthy volunteers who were free of significant ocular and
systemic pathology and were between the ages of 25 and 52 years of
age with 20/16 or better acuity served as subjects. Two of the three
subjects had their ocular aberrations measured using a laboratory
Shack-Hartmann wavefront sensor. The third subject’s ocular ab-
erration was not measured. The coefficients for a normalized
Zernike expansion through the 10th order for the two measured
subjects are provided in Tables 1 and 2 along with their associated
phase transfer functions (see Figs. 2 and 3).
Generation of Aberrated Acuity Charts
As in our previous work,2, 3 a commercially available program
called CTView was used to generate both high- and low-contrast
aberrated and unaberrated log MAR acuity charts. CTView introduces the aberration into the charts by directly setting the wavefront
Zernike coefficient values and performing a convolution of the resulting point-spread function with an image of an acuity chart. Each chart
was created using a random selection of letters from an equally identifiable letter set, scaled for a 10-ft test distance. Charts were printed on
8.5- ⫻ 11-in sheets of photographic grade paper using a high-resolution printer (600 dpi). High- and low- (11%) contrast charts were
generated 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 RMS
wavefront error was individually loaded into each Zernike mode for
0
2
⫺3
⫺1
1
3
the second (共C⫺2
2 , C2, C2兲), third (共C3 , C3 , C3, C3兲), and fourth
⫺4
⫺2
0
2
4
radial order (共C4 , C4 , C4, C4, C4兲) of the normalized Zernike expansion. The acuity measured for the zero RMS wavefront error conditions (12) for each subject served as the normalizing reference for
that subject (see below). Thus, a total of 144 different charts (6 levels
of RMS error ⫻ 12 Zernike modes ⫻ 2 contrast levels) were used in
the study.
For clinical reference, an equivalent spherical defocus for a given
RMS wavefront error can be calculated using the following general
formula for calculating sphere (S), cylinder (C), and axis (␣) from
Optometry and Vision Science, Vol. 80, No. 2, February 2003
Visual Acuity and Zernike Aberrations—Applegate et al.
TABLE 1.
Average normalized Zernike expansion coefficients for the
higher-order (third through tenth) aberrations over a 3-mm
pupil for subject RAA from five Hartmann-Shack wavefront
measurementa
n
m
␮m
n
m
3
3
3
3
4
4
4
4
4
5
5
5
5
5
5
6
6
6
6
6
6
6
7
7
7
7
7
7
7
7
⫺3
⫺1
1
3
⫺4
⫺2
0
2
4
⫺5
⫺3
⫺1
1
3
5
⫺6
⫺4
⫺2
0
2
4
6
⫺7
⫺5
⫺3
⫺1
1
3
5
7
⫺5.78E⫺02
2.13E⫺02
3.82E⫺03
⫺2.04E⫺02
3.39E⫺03
4.67E⫺04
3.30E⫺03
1.12E⫺03
3.43E⫺03
⫺9.91E⫺04
5.23E⫺03
⫺2.99E⫺03
⫺4.63E⫺04
2.11E⫺03
8.89E⫺04
1.13E⫺03
⫺4.80E⫺04
1.13E⫺04
4.04E⫺04
4.05E⫺04
⫺7.08E⫺04
⫺1.33E⫺03
1.04E⫺04
5.09E⫺05
⫺2.47E⫺04
1.02E⫺04
1.12E⫺05
⫺9.02E⫺05
⫺1.11E⫺04
2.92E⫺04
8
8
8
8
8
8
8
8
8
9
9
9
9
9
9
9
9
9
9
10
10
10
10
10
10
10
10
10
10
10
⫺8
⫺6
⫺4
⫺2
0
2
4
6
8
⫺9
⫺7
⫺5
⫺3
⫺1
1
3
5
7
9
10
⫺8
⫺6
⫺4
⫺2
0
2
4
6
8
10
␮m
4.93E⫺05
⫺5.97E⫺05
3.20E⫺05
1.75E⫺06
⫺1.27E⫺05
⫺1.80E⫺05
4.17E⫺05
6.70E⫺05
⫺7.41E⫺05
⫺3.21E⫺06
⫺1.76E⫺06
⫺1.09E⫺06
4.28E⫺06
⫺1.03E⫺06
⫺2.33E⫺07
1.27E⫺06
2.82E⫺06
⫺4.87E⫺06
2.65E⫺06
⫺5.81E⫺07
⫺6.92E⫺07
1.09E⫺06
⫺6.72E⫺07
⫺1.44E⫺07
1.79E⫺08
3.38E⫺07
⫺7.93E⫺07
⫺1.08E⫺06
1.15E⫺06
⫺1.58E⫺07
99
TABLE 2.
Average normalized Zernike expansion coefficients for the
higher-order (third through 10th) aberrations over a 3-mm
pupil for subject HG from five Hartmann-Shack wavefront
measurementa
n
m
␮m
n
m
␮m
3
3
3
3
4
4
4
4
4
5
5
5
5
5
5
6
6
6
6
6
6
6
7
7
7
7
7
7
7
7
⫺3
⫺1
1
3
⫺4
⫺2
0
2
4
⫺5
⫺3
⫺1
1
3
5
⫺6
⫺4
⫺2
0
2
4
6
⫺7
⫺5
⫺3
⫺1
1
3
5
7
⫺3.55E⫺02
1.96E⫺03
1.17E⫺02
9.79E⫺03
1.68E⫺03
8.39E⫺03
7.50E⫺03
⫺2.59E⫺03
⫺1.68E⫺03
1.83E⫺03
3.06E⫺03
1.20E⫺03
⫺3.96E⫺04
⫺1.15E⫺03
⫺2.10E⫺03
⫺9.24E⫺04
⫺2.89E⫺04
⫺8.91E⫺04
1.06E⫺03
1.34E⫺06
2.04E⫺04
⫺7.11E⫺04
9.78E⫺05
⫺5.65E⫺05
⫺1.64E⫺04
⫺2.28E⫺05
3.48E⫺05
7.98E⫺05
5.83E⫺05
⫺2.18E⫺06
8
8
8
8
8
8
8
8
8
9
9
9
9
9
9
9
9
9
9
10
10
10
10
10
10
10
10
10
10
10
⫺8
⫺6
⫺4
⫺2
0
2
4
6
8
⫺9
⫺7
⫺5
⫺3
⫺1
1
3
5
7
9
10
⫺8
⫺6
⫺4
⫺2
0
2
4
6
8
10
3.92E⫺06
4.55E⫺05
5.82E⫺06
4.54E⫺05
⫺2.90E⫺05
⫺5.20E⫺06
⫺7.26E⫺06
4.05E⫺05
⫺5.09E⫺05
2.43E⫺06
⫺1.26E⫺06
5.86E⫺08
2.85E⫺06
⫺4.00E⫺08
⫺9.00E⫺07
⫺1.95E⫺06
⫺2.63E⫺07
⫺1.27E⫺06
4.68E⫺08
1.21E⫺07
⫺5.48E⫺08
⫺8.44E⫺07
7.40E⫺08
⫺8.65E⫺07
1.19E⫺07
2.56E⫺07
5.82E⫺08
⫺7.56E⫺07
1.05E⫺06
⫺7.26E⫺07
Units are in ␮m; n is the radial order, and m is the angular
frequency of each Zernike mode. Total higher-order root mean
square error over a 3-mm pupil is 0.066 ␮m.
Units are in ␮m; n is the radial order, and m is the angular
frequency of each Zernike mode. Total higher-order root mean
square error over a 3-mm pupil is 0.041 ␮m.
the second radial order coefficients of the Zernike expansion fit to
wavefront error.
4 冑3C02
Equivalent defocus ⫽ ⫺
R2
a
2
4 冑3C02 ⫺ 2 冑6 冑共C22兲2 ⫹ 共C⫺2
2 兲
S⫽ ⫺
R2
4 冑6 冑共C22兲2 ⫹ 共C2⫺2兲2
C⫽ ⫺
R2
共
兲Ⲑ
␣ ⫽ tan⫺1 C2⫺2 Ⲑ C22 2
(1)
(2)
a
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 a
6-mm pupil corresponds to an equivalent defocus of 0.000, 0.038,
0.077, 0.115, 0.154, and 0.192 D, respectively. Chart appearance
for no wavefront error to 0.25 ␮m of RMS wavefront error loaded
into the defocus term (C02) is illustrated in Fig. 4.
(3)
In Equations 1 to 4, coefficients Cnm are Zernike mode coefficients expressed in micrometers. R is the radius of the pupil expressed in millimeters. S is the dioptric power of the sphere expressed 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 the
total RMS error is assumed to exist in the spherical defocus term (
C02), Equation 2 reduces to
Protocol
The optical quality of the normal eye is maximized when the eye
is optimally corrected using a traditional spherocylindrical correction, and the pupil is limited to approximately 3 mm.4 Stated
differently, a 3-mm artificial pupil optimizes the normal eye’s optical quality by balancing diffraction effects that result with smaller
pupils and higher-order aberrations that are passed with larger
pupils. To illustrate why it is important to optimize the optical
Optometry and Vision Science, Vol. 80, No. 2, February 2003
100
Visual Acuity and Zernike Aberrations—Applegate et al.
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/mm
corresponds to approximately 20/10 assuming a posterior nodal distance of 16.67 mm.
quality of the eye, Fig. 5 shows two-dimensional modulation transfer functions (MTF’s) for an eye with a 3-mm diffraction-limited
pupil (Fig. 5A); subject RAA’s eye with 0.066 ␮m of RMS over a
3-mm pupil (Fig. 5B); and in Fig. 5 C to H, the two dimensional
modulation 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 into
C02(defocus). Important for this experiment is to notice in Fig. 5
that 0.25 down to 0.15 ␮m of RMS decreases the MTF such that
a real eye viewing through a 3 mm pupil can see all relevant spatial
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/mm
corresponds to approximately 20/10 assuming a posterior nodal distance of 16.67 mm.
Optometry and Vision Science, Vol. 80, No. 2, February 2003
Visual Acuity and Zernike Aberrations—Applegate et al.
101
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 defocus
term (C02) 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. The
small bars at the side of each chart denote 0.0 logMAR (20/20). Aberrated charts were generated using CTView (www.sarverassociates.com).
frequencies (Fig. 5B). For the 0.10 RMS error condition, the real
eye’s 3-mm pupil passes approximately the same spatial
information.
Although each individual Zernike mode affects the two-dimensional MTF differently, modes near the center of the Zernike tree
adversely affect the MTF more than those near the edge. In this
study and in our prior work,2 defocus (C02), spherical aberration (
⫺2
C04), and secondary astigmatism (C⫺2
4 , C4 ) have the largest impact
on the MTF. The general rule illustrated in Fig. 5 for defocus holds
true for all Zernike modes tested. That is, the vast majority of the
spatial information contained in the charts are passed through the
3-mm pupil as long as the RMS error is ⱖ0.10 ␮m. Consequently,
one can examine the impact of 0.25 down to 0.10 ␮m of aberration over a 6-mm pupil with minimal to no loss in fidelity through
a 3-mm pupil. For the 0.05 test condition, the real eye’s 3-mm
condition will not pass the vast majority of the relevant spatial
information. Therefore, the results of the 0.05-␮m RMS wavefront error test condition are anticipated to be nearly equivalent to
the 0.00-␮m RMS normalizing condition. Said differently, the
3-mm pupil is now the limiting factor for the 0.05-␮m test condition and not the aberration placed in the chart.
To maximize the optical quality, the subject’s test eye was di-
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 pupil
for 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 Zenike
mode C02 (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 to
20/20 assuming a secondary nodal-to-retina distance of 16.67 mm. Color contour steps are in 0.1 increments. Two-dimensional modulation transfer
functions were created using CTView (Sarver and Associates, Celebration, FL, www.sarverassociates.com).
Optometry and Vision Science, Vol. 80, No. 2, February 2003
102
Visual Acuity and Zernike Aberrations—Applegate et al.
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.
lated with 1% cyclopentolate, and the subject viewed each of the
charts through a 3-mm artificial pupil aligned to the eye’s achromatic axis. Artificial pupil alignment was achieved using a foveal
achromatic alignicator5–7 and maintained using a bite bar
mounted to a three-dimensional vise. The fellow eye was occluded.
The test eye was refracted to maximum plus to best visual acuity for
the 10-ft test distance through the aligned 3-mm artificial pupil.
Each subject read each of the charts in random order three times
until five letters were missed. The total number of letters read
correctly up to the point of the fifth miss was recorded for each
chart. Chart illumination was maintained at 100 cd/m2.
RESULTS
To compare data across subjects, the data for each subject was
first normalized to the mean of the letters read correctly while
viewing the unaberrated chart.
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 linear
fit to letters lost as a function of RMS wavefront error (right). Error bars are ⫾1 SD.
Optometry and Vision Science, Vol. 80, No. 2, February 2003
Visual Acuity and Zernike Aberrations—Applegate et al.
L (gained or lost) ⫽ LC(A) ⫺ LC(UA)
(5)
where L ⫽ letters gained or lost
LC共A兲 ⫽ 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 letters
compared with the mean of the unaberrated condition. To illustrate, say that the total number of letters read correctly up to the
fifth miss under the unaberrated condition on the 36 trials averaged 64.67 letters read correctly (i.e., LC(UA) ⫽ 64.67). Now, say
that for the C02(defocus) ⫽ 0.25-␮m test condition, the number of
letters read correctly for the three trials were 58, 56, and 55. Using
Equation 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 the
number of letters read correctly for each test condition was small
for all subjects. To illustrate the precision with which the subjects
could read the aberrated and unaberrated charts, the average standard deviation in letters lost across all test conditions for the three
subjects were, 0.91 (RAA), 1.05 (HG), 1.18 (BB).
Consistent with our anticipation, the 0-␮m RMS error conditions and the 0.05-␮m RMS conditions were not significantly
different by paired t-test (p ⫽ 0.05). All other RMS levels were
significantly different from 0 RMS conditions and from each other
at p ⬍ 0.01 (Fig. 6 and Table 3). The two most probable explanations for the finding that the 0-␮m RMS and 0.05-␮m RMS
conditions were not significantly different are (1) the chart contained more spatial information than can pass through a 3-mm
artificial pupil (as discussed above) and (2) very low levels of RMS
error make no significant difference in our ability to measure highcontrast visual acuity. We include the RMS 0.05-␮m data in the
linear regressions below; it should be remembered that that this
condition is essentially equivalent to the normalizing 0-␮m RMS
error condition.
TABLE 3.
Paired t-test comparing the differences in means for all six
levels of root mean square wavefront error tested
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
00,
00,
00,
00,
00,
05,
05,
05,
05,
10,
10,
10,
15,
15,
20,
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
05a
10
15
20
25
10
15
20
25
15
20
25
20
25
25
Mean Difference
DF
t Value
p Value
0.556
1.500
2.741
4.315
5.704
0.944
2.185
3.759
5.148
1.241
2.815
4.204
1.574
2.963
1.389
107
107
107
107
107
107
107
107
107
107
107
107
107
107
107
1.741
4.820
9.046
11.924
16.337
3.348
7.661
11.244
15.618
4.699
8.808
15.034
5.120
9.622
4.671
0.0846
⬍0.0001
⬍0.0001
⬍0.0001
⬍0.0001
0.0011
⬍0.0001
⬍0.0001
⬍0.0001
⬍0.0001
⬍0.0001
⬍0.0001
⬍0.0001
⬍0.0001
⬍0.0001
C 00 ⫽ the 0.00-␮m wavefront error condition, C 05 ⫽ the
0.05-␮m wavefront error condition, etc.
a
103
The mean number of letters gained or lost for each subject for
each test condition was averaged across subjects to determine the
mean of means for each test condition. For each Zernike mode, the
mean of means was, in turn, plotted as a function of the RMS
wavefront error.
Fig. 7 illustrates the experimental Zernike mode yielding the
best (Fig. 7, left) and worst (Fig. 7, right) linear fit to the data.
Intercepts, slopes, and r2 are displayed in a Zernike pyramid for the
high- (Fig. 8) and low- (Fig. 9) contrast charts, respectively. Notice
in Figs. 8 and 9 that Zernike modes near the center of the pyramid
have the greatest impact on acuity as a function of RMS error (the
steepest slopes).
DISCUSSION
The coefficients of the normalized Zernike expansion reveal
the relative contributions of each Zernike mode to the total
RMS error. They do not reveal their relative contribution to
visual performance as measured by high- or low-contrast log
MAR acuity. As RMS error increases by ⬎0.05 ␮m, high- and
low-contrast acuity decreases and is well described by a linear
function across Zernike modes (Fig. 6) or for any single Zernike
mode (Figs. 7, 8, and 9).
The actual degree of independence of each mode in the formation
of the retinal image is not revealed in this study because each mode was
tested independently. In fact, in studies we allude to in the discussion
of our earlier work2 and later reported (R. Applegate, paper presented
at Refr@ctive.online, Topogr@phy.online, @berrometry.online Annual Meeting, Milan, Italy, September 2002. CD of the meeting to be
distributed by EyeWorld in 2003), we show acuity data demonstrating that aberrations from different modes interact. As we stated in the
published paper,2 “When combined, Zernike modes can interact to
improve acuity despite the increase in total wavefront error. For example, spherical aberration and defocus can be combined in such a manner that the individual modes affect vision more than the combination. Likewise, astigmatism and secondary astigmatism can be
combined such that their summed effect on acuity is less than the
individual modes. Modes two radial orders apart (e.g., radial orders 2
and 4) and having the same angular frequency (e.g., 0, ⫺2, or 2) can
be combined such that the combined effect on acuity is less than the
individual effects.” Furthermore, Zernike modes within the same radial order can interact to decrease visual performance to a greater
extent than loading the same total RMS error into a single term (R.
Applegate, reported at Refr@ctive.online, Topogr@phy.online, @berrometry.online Annual Meeting, Milan, Italy, September 2002. CD
of the meeting to be distributed by EyeWorld in 2003; this work has
been accepted for publication).8
New metrics (besides RMS error) that better correlate with
visual performance need to be developed. To this end, Larry
Thibos’ laboratory, David Williams’ laboratory, and our laboratory have joined forces to explore a variety of optical metrics
and combined optical neural metrics that better correlate with
measured visual performance. Fortunately, this endeavor is
made easier because carefully measured wavefront error fundamentally defines the optical properties of the system. A direct
consequence is that wavefront error can be used to calculate
other optical metrics in the pupil, spatial, or frequency do-
Optometry and Vision Science, Vol. 80, No. 2, February 2003
104
Visual Acuity and Zernike Aberrations—Applegate et al.
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.
mains. Capitalizing on this fact, we are in the process of evaluating over 20 optical metrics calculated from the wavefront
error against the visual performance results reported here and
elsewhere2 and will be reporting these results in the near future.
The most promising appear to be metrics that include compensation for both the optical and neural transfer functions. We
look forward to completing these studies and reporting them in
the near future.
CONCLUSIONS
Increasing the RMS error within any single mode of the normalized Zernike expansion decreases visual acuity in a linear fashion. The slope of the best fitting linear equation varies with
Zernike mode. Slopes near the center of the Zernike pyramid are
steeper than those near the edge. Although the normalized Zernike
expansion parcels RMS error orthogonally, the resulting effects on
Optometry and Vision Science, Vol. 80, No. 2, February 2003
Visual Acuity and Zernike Aberrations—Applegate et al.
visual performance as measured by visual acuity are not orthogonal. New metrics of the combined effects of the optical and the
neural transfer functions that are predictive of visual performance
need to be developed.
2.
3.
4.
ACKNOWLEDGMENTS
This work was supported by National Eye Institute, National Institutes of
Health grant R01 08520 to RAA, National Eye Institute, National Institutes
of Health grant R44 EY 11681 to EJS, HEAF Funds awarded to RAA from
the University of Houston, The Visual Optics Institute at the College of Optometry, University of Houston, and unrestricted funds awarded to the University of Texas Health Science Center San Antonio, Department of Ophthalmology from Research to Prevent Blindness. Results of this study were first
reported by C. Ballentine, et al. at the Association for Research in Vision and
Ophthalmology, Ft. Lauderdale, FL, May 2002.
R. A. Applegate, E. J. Sarver, and C. A. Sarver have proprietary interest in
the program CTView.
Received May 6, 2002; revision received November 22, 2002.
5.
6.
7.
8.
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Raymond A. Applegate
College of Optometry
University of Houston
Houston, TX 77204-2020
e-mail: rapplegate@uh.edu