Journal of Vision (2004) 4, 329-351 http://journalofvision.

org/4/4/9/ 329

Accuracy and precision of objective refraction from

wavefront aberrations

Larry N. Thibos School of Optometry, Indiana University, Bloomington, IN, USA

Xin Hong School of Optometry, Indiana University, Bloomington, IN, USA

Arthur Bradley School of Optometry, Indiana University, Bloomington, IN, USA

College of Optometry, University of Houston, Houston,

Raymond A. Applegate TX, USA

We determined the accuracy and precision of 33 objective methods for predicting the results of conventional, sphero-
cylindrical refraction from wavefront aberrations in a large population of 200 eyes. Accuracy for predicting defocus (as
specified by the population mean error of prediction) varied from –0.50 D to +0.25 D across methods. Precision of these
estimates (as specified by 95% limits of agreement) ranged from 0.5 to 1.0 D. All methods except one accurately pre-
dicted astigmatism to within ±1/8D. Precision of astigmatism predictions was typically better than precision for predicting
defocus and many methods were better than 0.5D. Paraxial curvature matching of the wavefront aberration map was the
most accurate method for determining the spherical equivalent error whereas least-squares fitting of the wavefront was
one of the least accurate methods. We argue that this result was obtained because curvature matching is a biased
method that successfully predicts the biased endpoint stipulated by conventional refractions. Five methods emerged as
reasonably accurate and among the most precise. Three of these were based on pupil plane metrics and two were based
on image plane metrics. We argue that the accuracy of all methods might be improved by correcting for the systematic
bias reported in this study. However, caution is advised because some tasks, including conventional refraction of defocus,
require a biased metric whereas other tasks, such as refraction of astigmatism, are unbiased. We conclude that objective
methods of refraction based on wavefront aberration maps can accurately predict the results of subjective refraction and
may be more precise. If objective refractions are more precise than subjective refractions, then wavefront methods may
become the new gold standard for specifying conventional and/or optimal corrections of refractive errors.

Keywords: visual optics, optical aberrations, refraction, metrics of optical quality

objective refraction is to prescribe correcting lenses based

Introduction on Zernike coefficients of the second-order.
The purpose of a conventional, ophthalmic refraction Unfortunately, the problem is not solved so easily. Sev-
of the eye is to determine that combination of spherical eral studies have shown that eliminating the second-order
and cylindrical lenses which optimizes visual acuity for dis- Zernike aberrations does not necessarily optimize the sub-
tant objects. The underlying assumption of refraction is jective impression of best-focus nor the objective measure-
that visual acuity is maximized when the quality of the reti- ment of visual performance (Applegate, Ballentine, Gross,
nal image is maximized. Furthermore, it is commonly as- Sarver, & Sarver, 2003; Applegate, Marsack, Ramos, &
sumed that retinal image quality is maximized when the Sarver, 2003; Guirao & Williams, 2003; Thibos, Hong,
image is optimally focused. For these reasons, the endpoint Bradley, & Cheng, 2002). Eliminating second-order
of a subjective refraction is taken as an operational defini- Zernike aberrations is equivalent to minimizing the root
tion of the term “best correction” as applied to eyes. mean squared (RMS) wavefront error, but this minimiza-
This paper is concerned with the problem of objectively tion does not necessarily optimize the quality of the retinal
determining the best correction of an eye from measure- image (King, 1968; Mahajan, 1991). Thus a search has be-
ments of wavefront aberrations. Aberrometers measure all gun for alternative metrics of optical quality that are opti-
of the eye’s monochromatic aberrations and display the mized by subjective refraction when higher-order aberra-
result in the form of an aberration map that describes the tions are present.
variation in optical path length from source to retinal im- A variety of problems must be solved when converting
age through each point in the pupil. Zernike expansion of an aberration map into a prescription for corrective lenses
an aberration map includes the second order aberrations of or refractive surgery. One of the most important is a correc-
defocus and astigmatism. Thus, one obvious strategy for tion for the eye’s chromatic aberration. Objective aber-
doi:10.1167/4.4.9 Received September 28, 2003; published April 23, 2004 ISSN 1534-7362 © 2004 ARVO

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 Journal of Vision (2004) 4, 329-351 Thibos, Hong, Bradley & Applegate 330

rometers typically use infrared light, for which the eye has approximation would eliminate the bulk of defocus error
relatively low refractive power compared to visible light. by correcting the eye with a spherical lens of power M, the
Optical models of longitudinal chromatic aberration so-called spherical equivalent. Next, the eye’s astigmatism is
(Thibos, Ye, Zhang, & Bradley, 1992) can be extrapolated corrected with a cylindrical lens, followed by a fine-tuning
to estimate the difference in optical power of the eye be- of the spherical lens power if necessary. This is the basis of
tween the measurement wavelength and some visible wave- most of the methods described below.
length, but it is unclear what wavelength should be chosen A different kind of problem is to incorporate into the
as a reference for any given eye. Furthermore, since only method the refractionist’s rule “maximum plus to best vis-
one wavelength can be in-focus at a time, some method is ual acuity” (Borisch, 1970). According to this clinical
needed to factor in the relative contribution of all wave- maxim, the spherical refractive error of myopic eyes should
lengths, each with a different amount of defocus and a dif- be deliberately under-corrected. The amount of under-
ferent luminance, in order to objectively refract an eye for correction is not enough to diminish visual acuity, but it is
polychromatic objects. sufficient to minimize unnecessary accommodation and to
Another sticky problem is the lack of a universally- maximize the usable depth of focus (DOF) at distance and
accepted metric of image quality that could be used to es- near. These twin goals are achieved by prescribing a spheri-
tablish objectively the state of optimum-focus for an aber- cal lens power that is slightly less negative (in the case of
rated eye. One purpose of this paper is to describe a variety myopia) or slightly more positive (in the case of hyperopia)
of such metrics based on general principles described else- than the lens required to make the retina conjugate to in-
where (Cheng, Thibos, & Bradley, 2003; Williams, Apple- finity. Instead, the prescribed lens conjugates the retina
gate, & Thibos, 2004). Assuming that consensus agreement with a plane at the hyperfocal distance, which is the nearest
could be achieved for a metric of choice, one still needs to distance the retina can focus on without significantly reduc-
deal with the fact that identifying the best correction is a ing visual performance for a target located at infinity
multi-dimensional problem in optimization. Guirao & Wil- (Campbell, 1957). Consequently, the eye is left in a slightly
liams (Guirao & Williams, 2003) have described an itera- myopic state (Figure 1B), compared to an optimum correc-
tive method for finding the optimum sphere, cylinder and tion that would place the retina conjugate to infinity
axis parameters that optimize a metric of image quality. (Figure 1A). Note that the diagram in Figure 1 has been
Other possibilities include an objective version of the clini- simplified by assuming that any astigmatism has already
cal technique of refraction by successive elimination. A first been fully corrected using the appropriate cylindrical lens.
Yet another issue is the extent to which neural factors
need to be taken into account when converting an aberra-
tion map into a prescription. One such neural factor is the
angular sensitivity of cone photoreceptors (Enoch &
Lakshminarayanan, 1991) which is commonly modeled
optically by an apodization filter in the pupil plane (Bradley
& Thibos, 1995; Metcalf, 1965). Post-receptoral neural
processing of the retinal image affects the processing of
blurred retinal images in a manner that can be modeled as
a mathematical convolution of the optical point-spread
function with a neural point-spread function (Thibos &
Bradley, 1995). This too may be construed as a form of
apodization since the effect of the convolution will be to
attenuate the remote tails of a blurred point-spread func-
tion (PSF).
Recently Guirao and Williams (Guirao & Williams,
2003) described a variety of methods for quantifying the
optical quality of an eye based on (1) analysis of wavefront
aberrations using pupil-plane metrics and (2) analysis of
retinal image quality using image-plane metrics. They re-
ported that all five image plane metrics they considered
Figure 1. Two criteria for refracting the eye. (A) An optimum re- were more accurate than two pupil-plane metrics in predict-
fraction conjugates the retina with infinity. In this case the ideal ing the optimum subjective refraction for a polychromatic
correcting lens images infinity at the eye’s far point (•). (B) A target for a small population of 6 eyes. Further testing was
conventional refraction conjugates the fovea with the eye’s hy- done on a large population of 146 eyes for which aberra-
perfocal point (•), which lies closer to the eye by an amount tion data for a fixed, 5.7 mm pupil were available in the
equal to half the depth-of-field (DOF). In this case the correcting literature. Unfortunately, a variety of uncontrolled condi-
lens images infinity at a point (o) slightly beyond the eye’s far tions precluded strong conclusions from this large popula-
point and therefore the eye remains slightly myopic. tion (e.g. possible fluctuations of accommodation, un-

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 Journal of Vision (2004) 4, 329-351 Thibos, Hong, Bradley & Applegate 331

known pupil size during subjective refraction, binocular −c20 4 3

refractions that likely yielded sub-optimal acuity endpoints) M=
but nevertheless the authors found a close correlation be- r2
tween subjective and objective refractions computed from
−c22 2 6
image-plane metrics. Although visual performance during J0 = (1)
refraction presumably depended on some combination of r2
optical and neural factors, they found that optimizing the
optical image without considering neural factors led to ac- −c2−2 2 6
J 45 =
curate prediction of the outcome of subjective refraction. r2
However, no assessment of the precision of these predic-
tions was reported. where cnm is the nth order Zernike coefficient of meridional
The purpose of our study was to evaluate two general frequency m, and r is pupil radius. The power vector nota-
approaches to converting an aberration map into a conven- tion is a cross-cylinder convention that is easily transposed
tional sphero-cylindrical prescription. The first approach is into conventional minus-cylinder or plus-cylinder formats
a surface-fitting procedure designed to find the nearest used by clinicians (see equations 22, 23 of Thibos,
sphero-cylindrical approximation to the actual wavefront Wheeler, & Horner, 1997).
aberration map. The second approach involves a virtual Paraxial curvature matching
through-focus experiment in which the computer adds or Curvature is the property of wavefronts that determines
subtracts various amounts of spherical or cylindrical wave- how they focus. Thus, another reasonable way to fit an ar-
fronts to the aberration map until the optical quality of the bitrary wavefront with a quadratic surface is to match the
eye is maximized. Preliminary accounts of this work have curvature of the two surfaces at some reference point. A
been presented (Thibos, Bradley, & Applegate, 2002; Thi- variety of reference points could be selected, but the natural
bos, Hong, & Bradley, 2001). choice is the pupil center. Two surfaces that are tangent at
a point and have exactly the same curvature in every merid-
Methods ian are said to osculate. Thus, the surface we seek is the
osculating quadric. Fortunately, a closed-form solution exists
for the problem of deriving the power vector parameters of
Refraction based on the principle of equiva- the osculating quadratic from the Zernike coefficients of
lent quadratic the wavefront (Thibos et al., 2002). This solution is ob-
We define the equivalent quadratic of a wavefront aber- tained by computing the curvature at the origin of the
ration map as that quadratic (i.e. a sphero-cylindrical) sur- Zernike expansion of the Seidel formulae for defocus and
face which best represents the map. This idea of approxi- astigmatism. This process effectively collects all r2 terms
mating an arbitrary surface with an equivalent quadratic is from the various Zernike modes. We used the OSA defini-
a simple extension of the common ophthalmic technique tions of the Zernike polynomials, each of which has unit
of approximating a sphero-cylindrical surface with an variance over the unit circle (Thibos, Applegate, Schwieger-
equivalent sphere. Two methods for determining the ling, & Webb, 2000). The results given in Equation 2 are
equivalent quadratic from an aberration map are presented truncated at the sixth Zernike order but could be extended
next. to higher orders if warranted.

Least-squares fitting −c20 4 3 + c40 12 5 − c60 24 7 + ...

M=
One common way to fit an arbitrarily aberrated wave- r2
front with a quadratic surface is to minimize the sum of
squared deviations between the two surfaces. This least- −c22 2 6 + c42 6 10 − c62 12 14 + ...
J0 = (2)
squares fitting method is the basis for Zernike expansion of r2
wavefronts. Because the Zernike expansion employs an or-
thogonal set of basis functions, the least-squares solution is −c2−2 2 6 + c4−2 6 10 − c6−2 12 14 + ...
given by the second-order Zernike coefficients, regardless of J 45 =
r2
the values of the other coefficients. These second-order
Zernike coefficients can be converted to a sphero-cylindrical Refraction based on maximizing optical or
prescription in power vector notation using Equation 1, visual quality
An empirical way to determine the focus error of an
eye (with accommodation paralyzed) is to move an object
axially along the line-of-sight until the retinal image of that
object appears subjectively to be well focused. This proce-
dure is easily simulated mathematically by adding a spheri-
cal wavefront to the eye’s aberration map and then comput-

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 Journal of Vision (2004) 4, 329-351 Thibos, Hong, Bradley & Applegate 332

ing the retinal image using standard methods of Fourier wavefront quality, (2) retinal image quality for point ob-
optics as illustrated in Movie 1. The curvature of the added jects, and (3) retinal image quality for grating objects. Im-
wavefront can be systematically varied to simulate a plementation of image sharpness metrics for extended ob-
through-focus experiment that varies the optical quality of jects, such as a letter chart, (Fienup & Miller, 2003; Hult-
the eye+lens system over a range from good to bad. Given a gren, 1990) have been left for future work. Several of the
suitable metric of optical quality, this computational pro- implemented metrics include a neural component that
cedure yields the optimum power M of that spherical cor- takes into account the spatial filtering of the retinal image
recting lens needed to maximize optical quality of the cor- imposed by the observer’s visual system. Strictly speaking,
rected eye. With this virtual spherical lens in place, the such metrics should be referred to as metrics of neuro-
process can be repeated for through-astigmatism calcula- optical quality or visual quality, but for simplicity we use
tions to determine the optimum values of J0 and J45 needed the term “optical quality metric” generically. For each of
to maximize image quality. If necessary, a second iteration these 31 metrics we used the virtual refraction procedure
could be used to fine-tune results by repeating the above described above to determine (to the nearest 1/8 D) the
process with these virtual lenses in place. However, the values of M, J0 and J45 required to maximize the metric.
analysis reported below did not include a second iteration. These objective refractions were then compared with con-
ventional subjective refractions. A listing of acronyms for
the various refraction methods is given in Table 1.

Evaluation of methods for objective refrac-

tion
To judge the success of an objective method of refrac-
tion requires a gold standard for comparison. The most
clinically relevant choice is a subjective refraction per-
formed for Sloan letter charts illuminated by white light.
Accordingly, we evaluated our objective refractions against
the published results of the Indiana Aberration Study
(Thibos et al., 2002). That study yielded a database of aber-
ration maps for 200 eyes that were subjectively well-
corrected by clinical standards. The methodology employed
avoided the problems mentioned above that limited the
conclusions drawn by Guirao & Williams. A brief summary
of the experimental procedure used in the Indiana Aberra-
Movie 1. Dynamic simulation of the through-focus method of tion Study is given next.
objective refraction. To determine the optimum value M of a
spherical defocusing lens, a pre-determined sequence of M- Subjective refractions were performed to the nearest
values are used to modulate the wavefront map in the same way 0.25D on 200 normal, healthy eyes from 100 subjects using
that a real lens alters the eye’s wavefront aberration function. the standard optometric protocol of maximum plus to best
From the new aberration map we compute the retinal point- visual acuity. Accommodation was paralyzed with 1 drop of
spread function (PSF), optical transfer function (OTF), and retinal 0.5% cyclopentalate during the refraction. Optical calcula-
image of an eye chart. Scalar metrics of optical quality are used tions were performed for the fully dilated pupil, which var-
to optimize focus (M). The process is then repeated to optimize ied between 6-9 mm for different eyes. The refractive cor-
astigmatism parameters J0, J45. This example is for an eye with rection was taken to be that sphero-cylindrical lens combi-
0.1 µm of spherical aberration. nation which optimally corrected astigmatism and conju-
gated the retina with the eye’s hyperfocal point (Figure 1b).
The computational method described above captures This prescribed refraction was then implemented with trial
the essence of clinical refraction by mathematically simulat- lenses and worn by the subject during subsequent aber-
ing the effects of sphero-cylindrical lenses of various pow- rometry (λ = 633 nm). This experimental design empha-
ers. Our method is somewhat simpler to implement than sized the effects of higher-order aberrations by minimizing
that described by Guirao & Williams (Guirao & Williams, the presence of uncorrected second-order aberrations. The
2003) who used an iterative searching method to determine eye’s longitudinal chromatic aberration was taken into ac-
that combination of spherical and cylindrical lenses which count by the different working distances used for aber-
maximizes the eye’s optical quality. Regardless of which rometry and subjective refraction as illustrated in Figure 2.
searching algorithm is used, a suitable metric of optical Assuming the eye was well focused for 570 nm when view-
quality is required as a merit function. Guirao and Wil- ing the polychromatic eye chart at 4 m, the eye would also
liams used 5 such metrics of image quality. In Appendix A have been focused at infinity for the 633nm laser light used
we expand their list to 31 metrics by systematically pursuing for aberrometry (Thibos et al., 1992).
three general approaches to quantifying optical quality: (1)

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 Journal of Vision (2004) 4, 329-351 Thibos, Hong, Bradley & Applegate 333

N Acronym Brief Description (Figure 3). Accuracy for the spherical component of refrac-
1 RMSw Standard deviation of wavefront tion was computed as the population mean of M as deter-
2 PV Peak-valley mined from objective refractions. Accuracy for the astig-
3 RMSs RMSs: std(slope)
4 PFWc Pupil fraction for wavefront (critical pupil)
matic component of refraction was computed as the popu-
5 PFWt Pupil fraction for wavefront (tessellation) lation mean of (Bullimore, Fusaro, & Adams, 1998) vec-
6 PFSt Pupil fraction for slope (tessellation) tors. Precision is a measure of the variability in results and
7 PFSc Pupil fraction for slope (critical pupil) is defined for M as twice the standard deviation of the
8 Bave Average Blur Strength population values, which corresponds to the 95% limits of
9 PFCt Pupil fraction for curvature (tessellation) agreement (LOA) (Bland & Altman, 1986). The confidence
10 PFCc Pupil fraction for curvature (critical pupil)
11 D50 50% width (min)
region for astigmatism is an ellipse computed for the
12 EW Equivalent width (min) bivariate distribution of J0 and J45. This suggests a definition
13 SM Sqrt(2nd moment) (min) of precision as the geometric mean of the major and minor
14 HWHH Half width at half height (arcmin) axes of the 95% confidence ellipse.
15 CW Correlation width (min) In our view, accuracy and precision are equally impor-
16 SRX Strehl ratio in space domain tant for refraction. A method that is precise but not accu-
17 LIB Light in the bucket (norm)
rate will yield the same wrong answer every time. Con-
18 STD Standard deviation of intensity (norm)
19 ENT Entropy (bits) versely, a method that is accurate but not precise gives dif-
20 NS Neural sharpness (norm) ferent answers every time and is correct only on average.
21 VSX Visual Strehl in space domain Thus we seek a method that is both accurate and precise.
22 SFcMTF Cutoff spat. freq. for rMTF (c/d) However, one might argue that lack of accuracy implies a
23 AreaMTF Area of visibility for rMTF (norm) systematic bias that could be removed by a suitable correc-
24 SFcOTF Cutoff spat. freq. for rOTF (c/d) tion factor applied to any individual eye. One way to obtain
25 AreaOTF Area of visibility for rOTF (norm)
26 SROTF Strehl ratio for OTF such a correction factor is to examine the population statis-
27 VOTF OTF vol/ MTF vol tics of a large number of eyes, as we have done in this
28 VSOTF Visual Strehl ratio for OTF study. Any systematic bias obtained for this group could
29 VNOTF CS*OTF vol/ CS*MTF vol then be used as a correction factor for future refractions,
30 SRMTF Strehl ratio for MTF assuming of course that the individual in question is well
31 VSMTF Visual Strehl ratio for MTF represented by the population used to determine the cor-
32 LSq Least squares fit
33 Curve Curvature fit
rection factor. Although this may be an expedient solution

Table 1. Listing of acronyms for refraction methods. Ordering is

that used in correlation matrices (Figures 8, A8).

633 nm

570 nm
4 meters

Figure 2. Schematic diagram of optical condition of the Indiana

Aberration Study. Yellow light with 570 nm wavelength is as-
sumed to be in focus during subjective refraction with a white-
light target at 4 m. At the same time, 633 nm light from a target
at infinity would be well focused because of the eye’s longitudinal
chromatic aberration.

Since all eyes were corrected with spectacle lenses dur- Figure 3. Graphical depiction of the concepts of precision and
ing aberrometry, the predicted refraction was M = J0 = J45 = accuracy as applied to the 1-dimensional problem of estimating
0. The level of success achieved by the 33 methods of objec- spherical power (left column of diagrams) and the 2-dimensional
tive refraction described above was judged on the basis of problem of estimating astigmatism (right column of diagrams).
precision and accuracy at matching these predictions

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 Journal of Vision (2004) 4, 329-351 Thibos, Hong, Bradley & Applegate 334

to the problem of objective refraction, it lacks the power of Refraction based on maximizing optical or
a theoretically sound account of the reasons for systematic visual quality
biases in the various metrics of optical quality.
Computer simulation of through-focus experiments to
determine that lens (either spherical or astigmatic) which
Results optimizes image quality are computationally intensive, pro-
ducing many intermediate results of interest but too volu-
minous to present here. One example of the type of inter-
Refraction based on equivalent quadratic mediate results obtained when optimizing the pupil frac-
The two methods for determining the equivalent quad- tion metric PFWc (see Table 1 for a list of acronyms) is
ratic surface for a wavefront aberration map gave consis- shown in Figure 6A. For each lens power over the range –1
tently different results. A frequency histogram of results for to +1 D (in 0.125 D steps) a curve is generated relating
the least-squares method (Figure 4A) indicated an average RMS wavefront error to pupil radius. Each of these curves
spherical refractive error of M = –0.39 D. In other words, crosses the criterion level (λ/4 in our calculations) at some
this objective method predicted the eyes were, on average, radius value. That radius is interpreted as the critical radius
significantly myopic compared to subjective refraction. To since it is the largest radius for which the eye’s optical qual-
the contrary, the method based on paraxial curvature ity is reasonably good. The set of critical radius values can
matching (Figure 4B) predicted an average refractive error then be plotted as a function of defocus, as shown in Figure
close to zero for our population. Both methods accurately 6B. This through-focus function peaks at some value of
predicted the expected astigmatic refraction as shown by
the scatter plots and 95% confidence ellipses in Figure 5.

J45
Number of eyes

J45

J0

Refractive error K (D) Figure 5. Scatter plots of (A) the least-squares fit of the wave-
front over the entire pupil and (B) paraxial curvature matching
methods of determining the two components of astigmatism.
Figure 4. Frequency distribution of results for the least-squares
Circles show the results for individual eyes, green cross indi-
method for fitting the wavefront aberration map with a quadratic
cates the mean of the 2-dimensional distribution, and ellipses are
surface. Dilated pupil size ranged from 6 to 9 mm across the
95% confidence intervals. Precision is the geometrical mean of
population.
the major and minor axes of the ellipse.

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 Journal of Vision (2004) 4, 329-351 Thibos, Hong, Bradley & Applegate 335

The accuracy and precision of the 31 methods for ob-

jective refraction based on optimizing metrics of optical
RMS wavefront error (µm)

quality, plus the two methods based on wavefront fitting,

are displayed in rank order in Figure 7. Mean accuracy var-
ied from –0.50 D to +0.25 D. The 14 most accurate meth-
ods predicted M to within 1/8 D and 24 methods were
accurate to within 1/4 D. The method of paraxial curvature
matching was the most accurate method, closely followed
by the through-focus method for maximizing the wavefront
quality metrics PFWc and PFCt. Least-squares fitting was
one of the least accurate methods (mean error = -0.39 D).
Precision of estimates of M ranged from 0.5 to 1.0 D.
A value of 0.5 D means that the error in predicting M for
95 percent of the eyes in our study fell inside the confi-
Normalized pupil radius dence range given by the mean ± 0.5 D. The most precise
method was PFSc (±0.49D), which was statistically signifi-

Mean ±SD (n=200)

Critical pupil dia (mm)

1 Curvature
PFWc
PFCt
SFcMTF
Spherical 5 LIB
refractive VSX
SFcOTF
error CW
EW
10 SRX
VS(MTF
NS
VOTF
PFSc
15 VNOTF
Rank

Additional defocus (D) areaMTF

STD
VSOTF
Figure 6. An example of intermediate results for the through- SROTF
20 HWHH
focus calculations needed to optimize the pupil fraction metric
PFSt
PFWc. (A) The RMS value is computed as a function of pupil areaOTF
radius for a series of defocus values added to the wavefront ab- PFCc
erration function of this eye. The pupil size at the intersection SRMTF
25 D50
points of each curve with the criterion level of RMS are plotted as PFWt
a function of lens power in (B). The optimum correcting lens for ENT
this eye is the added spherical power that maximized the critical RMSw
Least sq.
pupil diameter (and therefore maximized PFWc) which in this 30 RMSs
example is +0.125 D. SM
PV
Bave
defocus, which is taken as the optimum lens for this eye
using this metric. In this way the full dataset of Figure 6 is -1 -0.5 0 0.5
reduced to a single number.
Similar calculations were then repeated for other eyes Predicted Spherical Error (D)
in the population to yield 200 estimates of the refractive
error using this particular metric. A frequency histogram of Figure 7. Rank ordering (based on accuracy) of 33 methods for
these 200 values similar to those in Figure 4 was produced predicting spherical refractive error. Red symbols indicate means
for inspection by the experimenters. Such histograms were for metrics based on wavefront quality. Black symbols indicate
then summarized by a mean value, which we took to be a mean for metrics based on image quality. Error bars indicate ± 1
measure of accuracy, and a standard deviation, which standard deviation of the population. Numerical data are given in
(when doubled) was taken as a measure of precision. Table 2.

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 Journal of Vision (2004) 4, 329-351 Thibos, Hong, Bradley & Applegate 336

cantly better than the others (F-test for equality of variance, rics for predicting astigmatism ranged from ±0.32D to
5% significance level). Precision of the next 14 methods in ±1.0D and the 15 best methods were better than ±0.5D.
rank ranged from ±0.58D to ±0.65D. These values were Rank ordering of all methods for predicting astigmatism is
statistically indistinguishable from each other. This list of given in Table 3.
the 15 most precise methods included several examples In comparing the precision for predicting defocus and
from each of the three categories of wavefront quality, astigmatism we found that 7 metrics were in the top-15 list
point-image quality, and grating-image quality. Rank order- for both types of prediction. Five of these were also accu-
ing of all methods for predicting defocus is given in Table rate to within 1/8 D for predicting both defocus and
2. astigmatism. Thus 5 metrics (PFSc, PFWc, VSMTF, NS,
A similar process was used to determine the accuracy and PFCt) emerged as reasonably accurate and among the
for estimating astigmatism. We found that all methods ex- most precise. Three of these successful metrics were pupil
cept one (PFCc) had a mean error across the population of plane metrics and two were image plane metrics. These re-
less than 1/8 D. This accuracy is the best we could rea- sults demonstrate that accurate predictions of subjective
sonably expect, given that the subjective refractions and the refractions are possible with pupil plane metrics. How-
virtual refractions used to predict subjective refractions ever, such metrics do not include the process of image
were both quantized at 1/8 D of cross-cylinder power. Pre- formation that occurs in the eye, a process that must influ-
cision of astigmatism predictions was typically better than ence subjective image quality. For this reason, image-
precision for predicting defocus. The precision of all met-

Accuracy Precision Accuracy Precision

Rank Metric Mean Metric 2xSTD Rank Metric Mean Metric 2xSTD
1 PFCc 0.2406 PFSc 0.4927 1 HWHH 0.0155 LSq 0.3235
2 Curv -0.006 AreaOTF 0.5803 2 LIB 0.0164 PFSc 0.3315
3 PFWc -0.0063 VSOTF 0.5806 3 PFCt 0.0192 Bave 0.3325
4 PFCt -0.0425 PFWc 0.5839 4 AreaMTF 0.0258 RMSs 0.3408
5 SFcMTF -0.0425 LIB 0.5951 5 ENT 0.0273 RMSw 0.3429
6 LIB -0.0681 NS 0.5961 6 NS 0.0281 Curv 0.3568
7 VSX -0.0731 VSMTF 0.5987 7 VSX 0.03 PFWc 0.3639
8 SFcOTF -0.0737 EW 0.6081 8 PFSt 0.0305 PV 0.4278
9 CW -0.0912 SRX 0.6081 9 AreaOTF 0.0313 VSMTF 0.4387
10 EW -0.1006 AreaMTF 0.6112 10 EW 0.0343 AreaMTF 0.4423
11 SRX -0.1006 PFCt 0.6213 11 SRX 0.0343 NS 0.4544
12 VSMTF -0.1131 STD 0.63 12 SRMTF 0.038 PFCt 0.4715
13 NS -0.1144 SFcMTF 0.6343 13 VSMTF 0.0407 STD 0.4752
14 VOTF -0.125 VSX 0.6391 14 STD 0.0422 PFWt 0.4923
15 PFSc -0.1281 D50 0.6498 15 CW 0.0576 SM 0.4967
16 VNOTF -0.1575 CW 0.6558 16 RMSs 0.0589 SRMTF 0.5069
17 AreaMTF -0.165 PFWt 0.6575 17 VSOTF 0.0594 EW 0.5181
18 STD -0.1656 PFSt 0.6577 18 PFSc 0.0608 SRX 0.5181
19 VSOTF -0.1794 RMSw 0.6702 19 D50 0.0665 CW 0.5287
20 SROTF -0.1875 SFcOTF 0.6786 20 SM 0.0668 LIB 0.535
21 HWHH -0.200 SRMTF 0.6888 21 Bave 0.0685 AreaOTF 0.5444
22 PFSt -0.2162 SROTF 0.69 22 SROTF 0.0724 SFcMTF 0.5659
23 AreaOTF -0.2269 ENT 0.6987 23 PFWc 0.0745 VSX 0.5813
24 SRMTF -0.2544 LSq 0.7062 24 VOTF 0.0787 VSOTF 0.6796
25 D50 -0.2825 HWHH 0.7115 25 LSq 0.0899 HWHH 0.6796
26 PFWt -0.3231 RMSs 0.7159 26 RMSw 0.0909 SROTF 0.7485
27 ENT -0.3638 Curv 0.7202 27 Curv 0.0913 PFSt 0.7555
28 RMSw -0.3831 SM 0.7315 28 PV 0.098 SFcOTF 0.7821
29 LSq -0.3906 VNOTF 0.7486 29 PFWt 0.1039 VNOTF 0.816
30 RMSs -0.425 Bave 0.7653 30 VNOTF 0.1059 D50 0.8416
31 SM -0.4319 PV 0.7725 31 SFcOTF 0.113 ENT 0.8751
32 PV -0.4494 VOTF 0.8403 32 SFcMTF 0.1218 VOTF 0.9461
33 Bave -0.4694 PFCc 0.9527 33 PFCc 0.8045 PFCc 1.0005
Table 2. Rank ordering of methods for predicting astigmatism
Table 3. Rank ordering of methods for predicting spherical parameters J0 and J45 jointly. Acronyms in red type are wave-
equivalent M based on accuracy and precision. Acronyms in red front quality methods. Brief descriptions of acronyms are given in
type are wavefront quality methods. Brief descriptions of acro- Table 1. Detailed descriptions are in Appendix. Units are diop-
nyms are given in Table 1. Detailed descriptions are in Appendix. ters.
Units are diopters.

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 Journal of Vision (2004) 4, 329-351 Thibos, Hong, Bradley & Applegate 337

plane metrics of visual quality are more germane to vision Another interesting feature of Figure 8 is that some re-
models of the refraction process that seek to capture the fraction methods (e.g. PFCc, VOTF, VNOTF) are very
subjective notion of a well-focused retinal image poorly correlated with all other methods. This result for
(Williams, Applegate, & Thibos, 2004). metric PFCc is explained by the fact that PFCc was the only
metric to produce hyperopic refractions in the vicinity of
Correlation between multiple objective re- M=+0.25D. However, this argument does not apply to the
fractions for the same eye other two examples that are poorly correlated with most
other metrics even though these other metrics produced
One implication of the results presented above is that similar refractions on average (e.g. 20 (NS), 7 (PFSc), and
different methods of objective refraction that yield similar 23 (AreaMTF)). This result suggests that maximizing met-
refractions on average are likely to be statistically correlated. rics VOTF and VNOTF optimizes a unique aspect of opti-
We tested this prediction by computing the correlation cal and visual quality that is missed by other metrics. In
coefficient between all possible pairs of methods for pre- fact, these two metrics were specifically designed to capture
dicting M. The resulting correlation matrix is visualized in infidelity of spatial phase in the retinal image.
Figure 8. For example, the left-most column of tiles in the
matrix represents the Pearson correlation coefficient r be-
tween the first objective refraction method in the list Discussion
(RMSw) and all other methods in the order specified in
Table 1. Notice that the values of M predicted by optimiz- The least-squares method for fitting an aberrated wave-
ing RMSw are highly correlated with the values returned by front with a spherical wavefront is the basis of Zernike ex-
methods 3 (RMSs), 8 (Bave), 19 (ENT), and 32 (least- pansion to determine the defocus coefficient. The failure of
squares fit). As predicted, all of these metrics are grouped at this method to accurately predict the results of subjective
the bottom of the ranking in Figure 7. To the contrary, refraction implies that the Zernike coefficient for defocus is
refractions using RMSw are poorly correlated with values an inaccurate indicator of the spherical equivalent of re-
returned by methods 4 (PFWc), 9 (PFCt), 21 (VSX), 24 fractive error determined by conventional subjective refrac-
(SFcOTF), and 33 (Curvature fit). All of these metrics are tions. On average, this metric predicted that eyes in our
grouped at the top of the ranking in Figure 7, which fur- study were myopic by -0.39D when in fact they were well
ther supports this connection between accuracy and corre- corrected.
lation. A similar analysis of the correlation matrix for To the contrary, matching paraxial curvature accurately
astigmatism parameters is not as informative because there predicted the results of subjective refraction. This method
was very little difference between the various methods for is closely related to the Seidel expansion of wavefronts be-
predicting J0 and J45. cause it isolates the purely parabolic (r2) term. It also corre-
sponds to a paraxial analysis since the r2 coefficient is zero
when the paraxial rays are well focused. Although this
method was one of the least accurate methods for predict-
ing astigmatism, it nevertheless was accurate to within
1/8D. The curvature method was one of the most precise
methods for predicting astigmatism but was significantly
less precise than some other methods for predicting defo-
Metric number

cus. For this reason it was eliminated from the list of 5

most precise and accurate methods.
Figure 7 may be interpreted as a table of correction fac-
tors that could potentially make all of the predictions of
defocus equally accurate. While this might seem a reason-
able approach to improving accuracy, it may prove cumber-
some in practice if future research should show that the
correction factors vary with pupil diameter, age, or other
conditions.
We do not know why the various metrics have different
amounts of systematic bias, but at least two possibilities
have already been mentioned. First, to undertake the data
analysis we needed to make an assumption about which
Metric number wavelength of light was well focused on the retina during
subjective refraction with a polychromatic stimulus. We
Figure 8. Correlation matrix for values of M determined by objec- chose 570 nm as our reference wavelength based on theo-
tive refraction. Metric number is given in Table 1. retical and experimental evidence (Charman & Tucker,
1978; Thibos & Bradley, 1999) but the actual value is un-

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 Journal of Vision (2004) 4, 329-351 Thibos, Hong, Bradley & Applegate 338

known. Changing this reference wavelength by just 20 nm compute the DOF from the wavefront aberration map for
to 550 nm would cause a 0.1 D shift in defocus, which is a individual eyes.
significant fraction of the differences in accuracy between A variety of other factors may also contribute to the
the various metrics. range of inaccuracies documented in Figure 7. For exam-
A second source of bias may be attributed to the differ- ple, all of the image quality metrics reported in this paper
ence between optimal and conventional refraction meth- are based on monochromatic light. Generalizing these met-
ods. The objective refraction procedures described in this rics to polychromatic light might improve the predictions
paper are designed to determine the optimum refraction of the subjective refraction. Inclusion of Stiles-Crawford
(Figure 1a) whereas the subjective refractions were conven- apodization in the calculations might also improve the pre-
tional (Figure 1b). The difference between the two end- dictions. Also, it may be unrealistic to think that a single
points is half the depth-of-focus (DOF) of the eye. The metric will adequately capture the multi-faceted notion of
DOF for subjects in the Indiana Aberration Study is un- best-focus. A multi-variate combination of metrics which
known, but we would anticipate a value of perhaps ±0.25D captures different aspects of optical image quality may yield
(Atchison, Charman, & Woods, 1997) which is about half better predictions (Williams et al., 2004). Those metrics
the total range of focus values spanned in Figure 7. Accord- that included a neural component were configured with the
ingly, we may account for the results in Figure 7 by suppos- same neural filter, when in fact different individuals are
ing that the curvature matching technique happens to lo- likely to have different neural filters. Furthermore, the
cate the far end of the DOF interval (which is located at characteristics of the neural filter are likely to depend on
optical infinity in a conventional refraction) whereas some stimulus conditions. Koomen et al. (Koomen, Scolnik, &
middle-ranking metric (such as VSOTF) locates the middle Tousey, 1951) and Charman et al. (Charman, Jennings, &
of the DOF, located at the hyperfocal distance. This infer- Whitefoot, 1978) found that pupil size affects subjective
ence is consistent with the fact that most eyes in the Indi- refraction differently under photopic and scotopic illumi-
ana Aberration Study had positive spherical aberration. nation. They suggested that this might be due to different
Such eyes have less optical power for paraxial rays than for neural filters operating at photopic and scotopic light lev-
marginal rays. Consequently, the retina will appear to be els. A change in neural bandwidth of these filters would
conjugate to a point that is beyond the hyperfocal point if alter the relative weighting given to low and high spatial
the analysis is confined to the paraxial rays. frequency components of the retinal image, thereby altering
The preceding arguments suggest that the superior ac- the optimum refraction. This idea suggests future ways to
curacy of the curvature method for determining the spheri- test the relative importance of the neural component of
cal equivalent of a conventional refraction is due to a bias metrics of visual quality described here.
in this method that favors the far end of the eye’s DOF. In Variability in the gold standard of subjective refraction
short, curvature matching (and several other metrics with similar is another likely source of disagreement between objective
accuracy) is a biased method that successfully predicts a biased and subjective refractions. The range of standard deviations
endpoint. By the same argument, the biased curvature for predicting M across all metrics was only 1/8 D (0.29-
method is not expected to predict astigmatism accurately 0.42 D), indicating that the precision of all metrics was
because conventional refractions are unbiased for astigma- much the same. This suggests that the precision of objective
tism. Although this line of reasoning explains why the par- refraction might be dominated by a single, underlying
axial curvature method will locate a point beyond the hy- source of variability. That source might in fact be variability
perfocal point, we lack a convincing argument for why the in the subjective refraction. Bullimore et al found that the
located point should lie specifically at infinity. Perhaps fu- 95% limit of agreement for repeatability of refraction is ±
ture experiments that include measurement of the DOF as 0.75D, which corresponds to a standard deviation of 0.375
well as the hyperfocal distance will clarify this issue and at D (Bullimore et al., 1998). If the same level of variability
the same time help identify objective methods for deter- were present in our subjective refractions, then uncertainty
mining the hyperfocal distance. in determining the best subjective correction would have
Pursuing the above line of reasoning suggests that some been the dominant source of error. It is possible, therefore,
metric near the bottom of the accuracy ranking, such as that all of our objective predictions are extremely precise
RMSw, locates the near end of the DOF. This accounting is but this precision is masked by imprecision of the gold
consistent with the findings of Guirao and Williams standard of subjective refraction. If so, then an objective
(Guirao & Williams, 2003) and of Cheng et al. (Cheng, wavefront analysis that accurately determines the hyperfocal
Bradley, & Thibos, 2004) that the optimum focus lies point and the DOF with reduced variability could become
somewhere between the more distant paraxial focus and the new gold standard of refraction.
the nearer RMS focus. Taken together, the least-squares
and curvature fitting methods would appear to locate the Comparison with companion studies
two ends of the DOF interval. While perhaps a mere coin- The metrics of image quality described in this paper
cidence, if this intriguing result could be substantiated have a potential utility beyond objective refraction. For ex-
theoretically then it might become a useful method to ample, Cheng et al. (Cheng et al., 2004) and Marsack et al.

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 Journal of Vision (2004) 4, 329-351 Thibos, Hong, Bradley & Applegate 339

(Applegate, Marsack, & Thibos, 2004) both used the same Metrics of wavefront quality
implementation of these metrics described below (see A perfect optical system has a flat wavefront aberration
Appendix) to predict the change in visual acuity produced map and therefore metrics of wavefront quality are de-
when selected, higher-order aberrations are introduced into signed to capture the idea of flatness. An aberration map is
an eye. The experimental design of the Cheng study was flat if its value is constant, or if its slope or curvature is zero
somewhat simpler in that monochromatic aberrations were across the entire pupil. Since a wavefront, its slope, and its
used to predict monochromatic visual performance, curvature each admits to a different optical interpretation,
whereas Marsack used monochromatic aberrations to pre- we sought meaningful scalar metrics based on all three: the
dict polychromatic performance. Nevertheless, both studies wavefront aberration map, the slope map, and the curva-
concluded that changes in visual acuity are accurately pre- ture map. Programs for computing the metrics were written
dicted by the pupil plane metric PFSt and by the image in Matlab (The Mathworks, Inc.) and tested against known
plane metric VSOTF. Furthermore, both studies concluded examples.
that three of the least accurate predictors were RMSw,
HWHH, and VOTF. In addition, the Cheng study demon- Flatness metrics
strated that, as expected, those metrics which accurately Wavefront error describes optical path differences across
predicted changes in visual acuity also predicted the lens the pupil that give rise to phase errors for light entering the
power which maximized acuity in a through-focus experi- eye through different parts of the pupil. These phase errors
ment. This was an important result because it established a produce interference effects that degrade the quality of the
tight link between variations in monochromatic acuity and retinal image. An example of a wave aberration map is
monochromatic refraction. shown in Figure A-1. Two common metrics of wavefront
The superior performance of metric VSOTF is also flatness follow.
consistent with the present study. This metric lies in the
middle of the accuracy ranking for predicting M in a con-
ventional refraction, which suggests that it would have ac-
curately predicted M in an optimum refraction. (This point
is illustrated graphically in Figure 5 of the Cheng et al. pa-
per.) Furthermore, present results show that VSOTF is one
of the most precise methods for estimating M, which sug-
gests it is very good at monitoring the level of defocus in
the retinal image for eyes with a wide variety of aberration
structures. It follows that this metric should also be very
good at tracking the loss of visual performance when im-
ages are blurred with controlled amounts of higher-order
aberrations, as shown by the Cheng and Marsack studies.
Lastly, the Cheng and Marsack studies rejected RMSw,
HWHH, and VOTF as being among the least predictive
metrics. All three of these metrics were among the least
precise metrics for predicting M in the present study. It is
reasonable to suppose that the high levels of variability as-
sociated with these metrics would have contributed to the
poor performance recorded in those companion studies.

Appendix
This appendix summarizes a variety of metrics of visual
quality of the eye. Several of these metrics are in common
use, whereas others are novel. In the present study these
metrics are used to estimate conventional refractions.
Other studies have used these same metrics to estimate best
focus for monochromatic letters (Cheng et al., 2004) and to
predict the change in visual acuity that results from the in-
troduction of controlled amounts of selected, higher-order
aberrations into polychromatic letters (Applegate et al.,
Figure A-1. A theoretical wavefront aberration map for 1µm RMS
2004).
of the third-order aberration coma over a 6mm pupil.

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 Journal of Vision (2004) 4, 329-351 Thibos, Hong, Bradley & Applegate 340

RMSw = root-mean-squared wavefront error computed front slopes may be interpreted as transverse ray aberrations
over the whole pupil (microns) that blur the image. These ray aberrations can be conven-
iently displayed as a vector field (lower right diagram). The
1 
0.5 base of each arrow in this plot marks the pupil location and
∫ ( )
RMS w = 
2
w(x, y) − w dxdy  (A1) the horizontal and vertical components of the arrow are
A  proportional to the partial derivatives of the wavefront
 pupil 
map. If the field of arrows is collapsed so that all the tails
superimpose, the tips of the arrows represent a spot dia-
where w(x,y) is the wavefront aberration function defined gram (lower right diagram) that approximates the system
over pupil coordinates x,y, A = pupil area, and the integra- point-spread function (PSF).
tion is performed over the domain of the entire pupil. The root-mean-squared value of a slope map is a meas-
Computationally, RMSw is just the standard deviation of ure of the spreading of light rays that blur the image in one
the values of wavefront error specified at various pupil loca- direction. The total RMS value computed for both slope
tions. maps taken together is thus a convenient metric of wave-
PV = peak-to-valley difference (microns) front quality that may be interpreted in terms of the size of
the spot diagram.
( )
PV = max w(x, y) − min w(x, y) ( ) (A2) RMSs = root-mean-squared wavefront slope computed
over the whole pupil (arcmin)
PV is the difference between the highest and lowest points
in the aberration map. 0.5
1 2 
Wavefront slope is a vector-valued function of pupil posi-
tion that requires two maps for display, as illustrated in

A
( )
∫ wx (x, y) − wx + 
 
Figure A-2. One map shows the slope in the horizontal (x) RMSs =  pupil  (A3)
direction and the other map shows the slope in the vertical 
( ) 
2
(y) direction. (Alternatively, a polar-coordinate scheme  wy (x, y) − wy dxdy 
would show the radial slope and tangential slope.) Wave-

Figure A-2. Slope maps (upper row) are the partial derivatives of the wavefront map in Figure A-1. Information contained in these two
maps is combined in the lower right diagram, which shows the magnitude and direction of ray aberrations at a regular grid of points in
the pupil. The ray aberrations, in turn, can be used to generate the spot diagram in lower left. (Note, overlapping points in this example
conceal the fact that there are as many points in the spot diagram as there are arrows in the ray aberration map.) Slopes are specified
in units of milliradians (1mrad = 3.44 arcmin).

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 Journal of Vision (2004) 4, 329-351 Thibos, Hong, Bradley & Applegate 341

where wx=dw/dx and wy=dw/dy are the partial spatial de- k1 (x, y) + k2 (x, y)
rivatives (i.e. slopes) of w(x,y) and A = pupil area. M (x, y) =
2 (A4)
Wavefront curvature describes focusing errors that blur
the image. To form a good image at some finite distance, G(x, y) = k1 (x, y) ⋅ k2 (x, y)
wavefront curvature must be the same everywhere across
the pupil. A perfectly flat wavefront will have zero curvature where the principal curvature maps k1(x,y), k2(x,y) are com-
everywhere, which corresponds to the formation of a per- puted from M and G using
fect image at infinity. Like wavefront slope, wavefront cur-
vature is a vector-valued function of position that requires
more than one map for display (Figure A-3). Curvature var- k1 , k2 = M (x, y) ± M 2 (x, y) − G(x, y) (A5)
ies not only with pupil position but also with orientation at
any given point on the wavefront. The Gaussian and mean curvature maps may be obtained
Fortunately, Euler’s classic formula of differential ge- from the spatial derivatives of the wavefront aberration
ometry assures us that the curvature in any meridian can be map using textbook formulas (Carmo, 1976).
inferred from the principal curvatures (i.e. curvatures in the Given the principal curvature maps, we can reduce the
orthogonal meridians of maximum and minimum curva- dimensionality of wavefront curvature by computing blur
ture) at the point in question (Carmo, 1976). The principal strength at every pupil location. The idea of blur strength is
curvatures at every point can be derived from maps of to think of the wavefront locally as a small piece of a quad-
mean curvature M(x,y) and Gaussian curvature G(x,y) as ratic surface for which a power vector representation can be
follows. computed (Thibos et al., 1997). A power vector P
(Bullimore et al., 1998) is a 3-dimensional vector whose
5 coordinates correspond to the spherical equivalent (M), the
Mean Curvature, M 3 Princ. Curvature, k1
4
normal component of astigmatism (J0) and the oblique
2 component of astigmatism (J45). Experiments have shown
1
3
that the length of the power vector, which is the definition
2 of blur strength, is a good scalar measure of the visual im-
0
1
pact of sphero-cylindrical blur (Raasch, 1995). Thus, a map
-1 of the length of the power-vector representation of a wave-
front at each point in the pupil may be called a blur-
0
-2
-1 strength map (Figure A-3).
To compute the blur-strength map we first use the
-3

principal curvature maps to compute the astigmatism map

Gauss Curvature, G 8 Princ. Curvature, k2 1 k1 (x, y) − k2 (x, y)
J (x, y) = (A6)
6 0 2
4
-1
and then combine the astigmatism map with the mean cur-
2
-2 vature map using the Pythagorean formula to produce a
-3
blur strength map
0
-4

-2 B(x, y) = M 2 (x, y) + J 2 (x, y) (A7)

-5

The spatial average of this blur strength map is a scalar

Astigmatism, J 1.6 Blur Stength, B
3.5 value that represents the average amount of focusing error
1.4 in the system that is responsible for image degradation,
3
1.2
2.5
Bave = average blur strength (diopters)
1
2
0.8

1
∫
1.5
Bave =
0.6
B(x, y) dxdy (A8)
0.4 1 pupil area
pupil
0.2 0.5

0
0
Pupil fraction metrics
In addition to the 4 metrics described above, another 6
Figure A-3. Curvature maps derived from the wavefront in Figure metrics of wavefront quality can be defined based on the
1. Calibration bars have units of diopters. concept of pupil fraction. Pupil fraction is defined as the

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 Journal of Vision (2004) 4, 329-351 Thibos, Hong, Bradley & Applegate 342

fraction of the pupil area for which the optical quality of PFCc = PFc when critical pupil is defined as the concentric
the eye is reasonably good (but not necessarily diffraction- area for which Bave < criterion (e.g. 0.25D)
limited). A large pupil fraction is desirable because it means The second general method for determining the area of
that most of the light entering the eye will contribute to a the good pupil is called the tessellation or whole pupil
good-quality retinal image. method. We imagine tessellating the entire pupil with small
sub-apertures (about 1% of pupil diameter) and then label-
Area of good pupil ing each sub-aperture as good or bad according to some
Pupil Fraction = (A9)
Total area of pupil criterion (Figure A-4, right-hand diagram). The total area of
all those sub-apertures labeled good defines the area of the
Two general methods for determining the area of the good pupil from which we compute pupil fraction as
good pupil are illustrated in Figure A-4. The first method,
called the critical pupil or central pupil method, examines
Area of good subapertures
the wavefront inside a sub-aperture that is concentric with PFt = (A11)
the eye’s pupil (Corbin, Klein, & van de Pol, 1999; Total area of pupil
Howland & Howland, 1977). We imagine starting with a As with the concentric metrics, implementation of
small sub-aperture where image quality is guaranteed to be Equation 11 requires criteria for deciding if the wavefront
good (i.e. diffraction-limited) and then expanding the aper- over a sub-aperture is good. For example, the criterion
ture until some criterion of wavefront quality is reached. could be based on the wavefront aberration function,
This endpoint is the critical diameter, which can be used to
compute the pupil fraction (critical pupil method) as fol- PFWt = PFt when a good sub-aperture satisfies the crite-
lows rion PV < criterion (e.g. λ/4)
Alternatively, the criterion could be based on wavefront
2
 critical diameter  slope,
PFc =   (A10)
 pupil diameter  PFSt = PFt when a good sub-aperture satisfies the crite-
rion horizontal slope and vertical slope are both < criterion
(e.g. 1 arcmin)
Or the criterion could be based on wavefront curvature as
summarized by blur strength,
PFCt = PFt when a good sub-aperture satisfies the crite-
rion B < criterion (e.g. 0.25D)

Metrics of image quality for point objects

A perfect optical system images a point object into a
compact, high-contrast retinal image as illustrated in Figure
A-5. The image of a point object is called a point-spread
function (PSF). The PSF is calculated as the squared magni-
Figure A-4. Pupil fraction method for specifying wavefront qual- tude of the inverse Fourier transform of the pupil function
ity. Red circle in the left diagram indicates the largest concentric P(x,y), defined as
sub-aperture for which the wavefront has reasonably good qual-
ity. White stars in the right diagram indicate subapertures for
which the wavefront has reasonably good quality.

To implement Equation 10 requires some criterion for

what is meant by good wavefront quality. For example, the
criterion could be based on the wavefront aberration map,
PFWc = PFc when critical pupil is defined as the concentric
area for which RMSw < criterion (e.g. λ/4)
Alternatively, the criterion for good wavefront quality could
be based on wavefront slope,
PFSc = PFc when critical pupil is defined as the concentric
area for which RMSs < criterion (e.g. 1 arcmin)
Or the criterion could be based on wavefront curvature as Figure A-5. Measures of image quality for point objects are
represented by the blur strength map, based on contrast and compactness of the image.

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 Journal of Vision (2004) 4, 329-351 Thibos, Hong, Bradley & Applegate 343

(
P(x, y) = A(x, y) exp ikW (x, y) ) (A12) 4 PSF(x, y)dxdy 
0.5

 ∫ 
pupil
where k is the wave number (2π/wavelength) and A(x,y) is EW =   (A14)
 π PSF(x0 , y0 ) 
an optional apodization function of pupil coordinates x,y.  
 
When computing the physical retinal image at the entrance
apertures of the cone photoreceptors, the apodization func-
tion is usually omitted. However, when computing the vis- where x0 , y0 are the coordinates of the peak of the PSF. In
ual effectiveness of the retinal image, the waveguide nature this and following equations, x,y are spatial coordinates of
of cones must be taken into account. These waveguide the retinal image, typically specified as visual angles sub-
properties cause the cones to be more sensitive to light en- tended at the eye’s nodal point. Note that although EW
tering the middle of the pupil than to light entering at the describes spatial compactness, it is computed from PSF
margin of the pupil (Burns, Wu, Delori, & Elsner, 1995; contrast. As the height falls the width must increase to
Roorda & Williams, 2002; Stiles & Crawford, 1933). It is maintain a constant volume under the PSF.
common practice to model this phenomenon as an apodiz- SM = square root of second moment of light distribution
ing filter with transmission A(x,y) in the pupil plane (arcmin)
(Atchison, Joblin, & Smith, 1998; Bradley & Thibos, 1995;
Zhang, Ye, Bradley, & Thibos, 1999). This metric is analogous to the moment of inertia of a
Scalar metrics of image quality that measure the quality distribution of mass. It is computed as
of the PSF in aberrated eyes are designed to capture the
0.5
dual attributes of compactness and contrast. The first 5  
metrics listed below measure spatial compactness and in
every case small values of the metric indicate a compact PSF  pupil
( )
 ∫ x 2 + y 2 PSF(x, y) dxdy 

SM =   (A15)
of good quality. The last 6 metrics measure contrast and in  
every case large values of the metric indicate a high-contrast 

∫ PSF(x, y) dxdy 

PSF of good quality. Most of the metrics are completely pupil
optical in character, but a few also include knowledge of
the neural component of the visual system. Several of these Unlike D50 above, this compactness metric is sensitive to
metrics are 2-dimensional extensions of textbook metrics the shape of the PSF tails. Large values of SM indicate a
defined for 1-dimensional impulse response functions rapid roll-off of the optical transfer function at low spatial
(Bracewell, 1978). Many of the metrics are normalized by frequencies (Bracewell, 1978).
diffraction-limited values and therefore are unitless.
HWHH = half width at half height (arcmin)
Compactness metrics
This metric is the average width of every cross-section
D50 = diameter of a circular area centered on PSF peak of the PSF. It is computed as
which captures 50% of the light energy (arcmin)
0.5
The value of D50 is equal to the radius r, where r is de- 1 
fined implicitly by: HWHH =  ∫ C(x, y) dxdy  (A16)
π 
 pupil 
2π r

∫ ∫ PSFN (r,θ )rdrdθ = 0.5 (A13) where C(x,y) = 1 if PSF(x,y) > max(PSF)/2, otherwise C(x,y)
0 0 = 0. A 1-dimensional version of this metric has been used
on line spread functions of the eye (Charman & Jennings,
where PSFN is the normalized (i.e. total intensity = 1) point- 1976; Westheimer & Campbell, 1962).
spread function with its peak value located at r = 0. This
metric ignores the light outside the central 50% region, CW = correlation width of light distribution (arcmin)
and thus is insensitive to the shape of the PSF tails. This metric is the HWHH of the autocorrelation of the
EW = equivalent width of centered PSF (arcmin) PSF. It is computed as
The equivalent width of the PSF is the diameter of the 0.5
circular base of that right cylinder which has the same vol- 1 ∞ ∞ 
ume as the PSF and the same height. The value of EW is CW = 
π
∫ ∫ Q(x, y) dxdy  (A17)
given by:  −∞ −∞ 

where Q(x,y) = 1 if PSF ⊗ PSF > max( PSF ⊗ PSF )/2,

otherwise Q(x,y) = 0. In this expression, PSF ⊗ PSF is the
autocorrelation of the PSF.

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 Journal of Vision (2004) 4, 329-351 Thibos, Hong, Bradley & Applegate 344

Contrast metrics NS = neural sharpness

SRX = Strehl ratio computed in spatial domain This metric was introduced by Williams as a way to
capture the effectiveness of a PSF for stimulating the neural
This widely-used metric is typically defined with respect portion of the visual system (Williams, 2003). This is
to the peak of the PSF, rather than the coordinate origin. It achieved by weighting the PSF with a spatial sensitivity
is computed as function that represents the neural visual system. The
product is then integrated over the domain of the PSF.
SRX =
( )
max PSF
(A18)
Here we normalize the result by the corresponding value
max (PSFDL ) for a diffraction-limited PSF to achieve a metric that is
analogous to the Strehl ratio computed for a neurally-
weighted PSF,
where PSFDL is the diffraction-limited PSF for the same
pupil diameter.
LIB = light-in-the-bucket ∫ PSF(x, y) g(x, y)dxdy
psf
The value of this metric is the percentage of total en- NS = (A22)
ergy falling in an area defined by the core of a diffraction-
limited PSF, ∫ PSFDL (x, y) g(x, y)dxdy
psf

where g(x,y) is a bivariate-Gaussian, neural weighting-

LIB = ∫ PSFN (x, y) dxdy (A19)
function. A profile of this weighting function (Figure A-6)
DL core
shows that it effectively ignores light outside of the central
4 arc minutes of the PSF.
where PSFN is the normalized (i.e. total intensity = 1) point-
spread function. The domain of integration is the central
core of a diffraction-limited PSF for the same pupil diame-
ter. An alternative domain of interest is the entrance aper-
ture of cone photoreceptors. Similar metrics have been
used in the study of depth-of-focus (Marcos, Moreno, &
Visual weight

Navarro, 1999).
STD = standard deviation of intensity values in the PSF,
normalized to diffraction-limited value
This metric measures the variability of intensities at
various points in the PSF,

0.5
 
( )
2 Visual angle (arcmin)
 dxdy 
∫ PSF(x, y) − PSF
 psf 
STD =   (A20) Figure A-6. Neural weighting functions used by NS and VSX.
0.5
 
( ) dxdy 
2

∫ PSFDL (x, y) − PSF DL
 psf
VSX = visual Strehl ratio computed in the spatial domain.
  Like the neural sharpness metric, the visual Strehl ratio
is an inner product of the PSF with a neural weighting
where PSFDL is the diffraction-limited point-spread func- function normalized to the diffraction-limited case. The
tion. The domain of integration is a circular area centered difference between NS and VSX is in the choice of weight-
on the PSF peak and large enough in diameter to capture ing functions (Figure A-6).
most of the light in the PSF.
ENT = entropy of the PSF
∫ PSF(x, y) N (x, y)dxdy
This metric is inspired by an information-theory ap- psf
VSX = (A23)
proach to optics (Guirao & Williams, 2003).
∫ PSFDL (x, y) N (x, y)dxdy
psf
ENT = − ∫ PSF(x, y) ln(PSF(x, y))dxdy (A21)
psf

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 Journal of Vision (2004) 4, 329-351 Thibos, Hong, Bradley & Applegate 345

where N(x,y) is a bivariate neural weighting function equal called a phase transfer function (PTF). Together, the MTF
to the inverse Fourier transform of the neural contrast sen- and PTF comprise the eye’s optical transfer function
sitivity function for interference fringes (Campbell & (OTF). The OTF is computed as the Fourier transform of
Green, 1965). With this metric, light outside of the central the PSF.
3 arc minutes of the PSF doubly detracts from image qual-
ity because it falls outside the central core and within an Optical theory tells us that any object can be conceived
inhibitory surround. This is especially so for light just out- as the sum of gratings of various spatial frequencies, con-
side of the central 3 arc minutes in that slightly aberrated trasts, phases and orientations. In this context we think of
rays falling 2 arc minutes away from the PSF center are the optical system of the eye as a filter that lowers the con-
more detrimental to image quality than highly aberrated trast and changes the relative position of each grating in the
rays falling farther from the center. object spectrum as it forms a degraded retinal image. A
high-quality OTF is therefore indicated by high MTF values
Metrics of image quality for grating objects and low PTF values. Scalar metrics of image quality in the
Unlike point objects, which can produce an infinite va- frequency domain are based on these two attributes of the
riety of PSF images depending on the nature of the eye’s OTF.
aberrations, small patches of grating objects always produce SFcMTF = spatial frequency cutoff of radially-averaged
sinusoidal images no matter how aberrated the eye. Conse- modulation-transfer function (rMTF)
quently, there are only two ways that aberrations can affect
Cutoff SF is defined here as the intersection of the ra-
the image of a grating patch: they can reduce the contrast
dially averaged MTF (rMTF) and the neural contrast
or translate the image sideways to produce a phase-shift, as
threshold function (Thibos, 1987). The rMTF is computed
illustrated in Figure A-7. In general, the amount of contrast
by integrating the full 2-dimensional MTF over orientation.
attenuation and the amount of phase shift both depend on
This metric does not capture spatial phase errors in the
the gratings spatial frequency. This variation of image con-
image because rMTF is not affected by the PTF portion of
trast with spatial frequency for an object with 100% con-
the OTF.
trast is called a modulation transfer function (MTF). The
variation of image phase shift with spatial frequency is
SFcMTF = highest spatial freq. for which
rMTF > neural threshold (A24)

where

2π

∫ abs (OTF( f , φ ))dφ

1
rMTF( f ) = (A25)
2π
0

and OTF(f,φ) is the optical transfer function for spatial fre-

quency coordinates f (frequency) and φ (orientation). A
graphical depiction of SFcMTF is shown in Figure A-8.
SFcOTF = spatial frequency cutoff of radially-averaged
optical-transfer function (rOTF).
The radially-averaged OTF is determined by integrating the
full 2-dimensional OTF over orientation. Since the PTF
component of the OTF is taken into account when com-
puting rOTF, this metric is intended to capture spatial
phase errors in the image.
Figure A-7. Measures of image quality for grating objects are SFcOTF = lowest spatial freq. for which
based on contrast and phase shifts in the image. Upper row de- rOTF < neural threshold (A26)
picts a high-quality image of a grating object. Lower row depicts
a low quality image with reduced contrast and a 180 deg phase where
shift. Left column shows the grating images and right column
show horizontal traces of intensity through the corresponding 2π
1
images. Red lines are reference marks that highlight the phase rOTF( f ) =
2π ∫ OTF( f , φ ) dφ .
shifts that can occur in blurred images. 0

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 Journal of Vision (2004) 4, 329-351 Thibos, Hong, Bradley & Applegate 346

The primary distinction between metrics SFcMTF and

SFcOTF is that SFcMTF ignores phase errors, with phase-
altered and even phase-reversed modulations treated the
same as correct-phase modulations. For example, with an
amplitude-oscillating and phase-reversing defocused OTF,
the SFcMTF identifies the highest frequency for which
modulation exceeds threshold, irrespective of lower fre-
quency modulation minima and phase reversals (Figure A-
8). By contrast, SFcOTF identifies the highest SF within the
correct-phase, low-frequency portion of the OTF (Figure A-
9). This allows spurious resolution to be discounted when
predicting visual performance on tasks of spatial resolution
and pattern discrimination.
AreaMTF = area of visibility for rMTF (normalized to dif-
fraction-limited case).
The area of visibility in this context is the region lying
below the radially averaged MTF and above the neural con-
trast threshold function (Charman & Olin, 1965; Mour-
Figure A-8. Radial MTF for a defocused optical system, showing oulis, 1999). The normalized metric is computed as
intersection with neural threshold function that defines cutoff
spatial frequency metric SFcMTF. Shaded area below the MTF cutoff cutoff
and above the neural threshold is the area of visibility specified
in the definition of metric AreaMTF.
∫ rMTF( f ) df − ∫ TN ( f ) df
AreaMTF = 0 0
(A27)
cutoff cutoff

and OTF(f,φ) is the optical transfer function for spatial fre- ∫ rMTFDL ( f ) df − ∫ TN ( f ) df
quency coordinates f (frequency) and φ (orientation). Since 0 0

the OTF is a complex-valued function, integration is per-

formed separately for the real and imaginary components. where TN is the neural contrast threshold function, which
Conjugate symmetry of the OTF ensures that the imaginary equals the inverse of the neural contrast sensitivity function
component vanishes, leaving a real-valued result. A graphi- (Campbell & Green, 1965). When computing area under
cal depiction of SFcOTF is shown in Figure A-9. rMTF, phase-reversed segments of the curve count as posi-
tive area (Figure A-8). This is consistent with our definition
of SFcMTF as the highest frequency for which rMTF ex-
ceeds neural theshold. This allows spurious resolution to be
counted as beneficial when predicting visual performance
for the task of contrast detection. Metrics based on the vol-
ume under the MTF have been used in studies of chro-
matic aberration (Marcos, Burns, Moreno-Barriusop, &
Navarro, 1999) and visual instrumentation (Mouroulis,
real(OTF)

1999).
AreaOTF = area of visibility for rOTF (normalized to dif-
fraction-limited case).
The area of visibility in this context is the region that
lies below the radially averaged OTF and above the neural
contrast threshold function. The normalized metric is
computed as
Spatial Frequency (cyc/deg) cutoff cutoff

∫ rOTF( f ) df − ∫ TN ( f ) df
Figure A-9. Radial OTF for a defocused optical system, showing AreaOTF = 0 0
(A28)
cutoff cutoff
intersection with neural threshold function to define cutoff spatial
frequency metric SFcOTF. Shaded area below the OTF and ∫ rOTFDL ( f ) df − ∫ TN ( f ) df
0 0
above the neural threshold is the area of visibility specified in the
definition of metric AreaOTF.

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 Journal of Vision (2004) 4, 329-351 Thibos, Hong, Bradley & Applegate 347

where TN is the neural contrast threshold function defined ∞ ∞

above. Since the domain of integration extends only to the ∫ ∫ OTF( f x , f y ) df x df y
cutoff spatial frequency, phase-reversed segments of the SROTF = −∞ −∞
(A30)
curve do not contribute to area under rOTF. This is consis- ∞ ∞
tent with our definition of SFcOTF as the lowest frequency ∫ ∫ OTFDL ( f x , f y ) df x df y
for which rOTF is below neural theshold. This metric −∞ −∞
would be appropriate for tasks in which phase reversed
modulations (spurious resolution) actively interfere with VSMTF = visual Strehl ratio computed in frequency do-
performance. main (MTF method)
SRMTF = Strehl ratio computed in frequency domain This metric is similar to the MTF method of comput-
(MTF method) ing the Strehl ratio, except that the MTF is weighted by the
The Strehl ratio is often computed in the frequency neural contrast sensitivity function CSFN,
domain on the strength of the central ordinate theorem of
Fourier analysis (Bracewell, 1978). This theorem states that ∞ ∞
the central value of a function is equal to the area (or vol- ∫ ∫ CSFN ( f x , f y ) ⋅ MTF( f x , f y ) df x df y
ume, in the 2-dimensional case) under its Fourier trans- VSMTF = −∞ −∞
(A31)
form. Since the OTF is the Fourier transform of the PSF, ∞ ∞

we may conclude that the volume under the OTF is equal ∫ ∫ CSFN ( f x , f y ) ⋅ MTFDL ( f x , f y ) df x df y
to the value of the PSF at the coordinate origin. In many −∞ −∞
cases the PTF portion of the OTF is unknown, which has
led to the popular substitution of the MTF for the OTF in In so doing, modulation in spatial frequencies above the
this calculation. Although popular, this method lacks rig- visual cut-off of about 60 c/deg is ignored, and modulation
orous justification because MTF=|OTF|. This non-linear near the peak of the CSF (e.g. 6 c/deg) is weighted maxi-
transformation destroys the Fourier-transform relationship mally. It is important to note that this metric gives weight
between the spatial and frequency domains that is the basis to visible, high spatial-frequencies employed in typical vis-
of the central ordinate theorem, which in turn is the justifi- ual acuity testing (e.g. 40 c/deg in 20/15 letters). Visual
cation for computing Strehl ratio in the frequency domain. Strehl ratio computed by the MTF method is equivalent to
the visual Strehl ratio for a hypothetical PSF that is well-
∞ ∞ centered with even symmetry computed as the inverse Fou-
∫ ∫ MTF( f x , f y ) df x df y rier transform of MTF (which implicitly assumes. PTF=0).
Thus, in general, VSMTF is only an approximation of the
−∞ −∞
SRMTF = (A29)
∞ ∞ visual Strehl ratio computed in the spatial domain (VSX).
∫ ∫ MTFDL ( f x , f y ) df x df y
VSOTF = visual Strehl ratio computed in frequency do-
−∞ −∞
main (OTF method)
Strehl ratio computed by the MTF method is equivalent to This metric is similar to the OTF method of computing
the Strehl ratio for a hypothetical PSF that is well-centered the Strehl ratio, except that the OTF is weighted by the
with even symmetry computed as the inverse Fourier trans- neural contrast sensitivity function CSFN,
form of MTF (which implicitly assumes. PTF=0). Thus, in
general, SRMTF is only an approximation of the actual ∞ ∞
Strehl ratio computed in the spatial domain (SRX). ∫ ∫ CSFN ( f x , f y ) ⋅ OTF( f x , f y ) df x df y
−∞ −∞
SROTF = Strehl ratio computed in frequency domain VSOTF = (A32)
∞ ∞
(OTF method)
∫ ∫ CSFN ( f x , f y ) ⋅ OTFDL ( f x , f y ) df x df y
The Strehl ratio computed by the OTF method will ac- −∞ −∞
curately compute the ratio of heights of the PSF and a dif-
fraction-limited PSF at the coordinate origin. However, the This metric differs from VSX by emphasizing image quality
peak of the PSF does not necessarily occur at the coordi- at the coordinate origin, rather than at the peak of the PSF.
nate origin established by the pupil function. Conse- VOTF = volume under OTF normalized by the volume un-
quently, the value of SROTF is not expected to equal SRX der MTF
exactly, except in those special cases where the peak of the
PSF occurs at the coordinate origin. This metric is intended to quantify phase shifts in the
image. It does so by comparing the volume under the OTF
to the volume under the MTF.

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 Journal of Vision (2004) 4, 329-351 Thibos, Hong, Bradley & Applegate 348

∞ ∞ Given this definition, PSFpoly may be substituted for PSF in

∫ ∫ OTF( f x , f y ) df x df y any of the equations given above to produce new, poly-
VOTF = −∞ −∞
(A33) chromatic metrics of image quality. In addition to these
∞ ∞ luminance metrics of image quality, other metrics can be
∫ ∫ MTF( f x , f y ) df x df y devised to capture the changes in color appearance of the
−∞ −∞ image caused by ocular aberrations. For example, the
chromaticity coordinates of a point source may be com-
Since the MTF ≥ the real part of the OTF, this ratio is al- pared to the chromaticity coordinates of each point in the
ways ≤ 1. Creation of this metric was inspired by a measure retinal PSF and metrics devised to summarize the differ-
of orientation bias of the receptive fields of retinal ganglion ences between image chromaticity and object chromaticity.
cells introduced by Thibos & Levick (Exp. Brain Research, Such metrics may prove useful in the study of color vision.
58:1-10, 1985). Given the polychromatic point-spread function defined
VNOTF = volume under neurally-weighted OTF, normal- above in Equation A-36, a polychromatic optical transfer
ized by the volume under neurally-weighted MTF function OTFpoly may be computed as the Fourier trans-
form of PSFpoly. Substituting this new function for OTF
This metric is intended to quantify the visually- (and its magnitude for MTF) in any of the equations given
significant phase shifts in the image. It does so by weighting above will produce new metrics of polychromatic image
the MTF and OTF by the neural contrast sensitivity func- quality defined in the frequency domain. Results obtained
tion before comparing the volume under the OTF to the by these polychromatic metrics will be described in a future
volume under the MTF. report.
∞ ∞
Correlation between metrics
∫ ∫ CSFN ( f x , f y ) ⋅ OTF( f x , f y ) df x df y Since all of the metrics of wavefront and image quality
−∞ −∞
VNOTF = (A34) defined above are intended to measure the optical quality
∞ ∞
of an eye, they are expected to be statistically correlated. To
∫ ∫ CSFN ( f x , f y ) ⋅ MTF( f x , f y ) df x df y estimate the degree of correlation for normal healthy eyes,
−∞ −∞
we computed each of these monochromatic, non-apodized
metrics for all 200 well refracted eyes of the Indiana Aber-
ration Study (Thibos, Hong, Bradley, & Cheng, 2002). The
Polychromatic metrics
correlation matrix for all 31 metrics of optical quality is
The wavefront aberration function is a monochromatic shown in Figure A-10. Only a few correlations were found
concept. If a source emits polychromatic light, then wave- to be statistically insignificant (α=0.05) and these were
front aberration maps for each wavelength are treated sepa- coded as zero in the figure.
rately because lights of different wavelengths are mutually Given the strong correlations between metrics evident
incoherent and do not interfere. For this reason, the defini- in Figure A-10, the question arose whether it would be pos-
tion of metrics of wavefront quality do not generalize easily sible to discover a smaller set of uncorrelated variables that
to handle polychromatic light. Nevertheless, it is possible to could adequately account for the individual variability in
compute the value of a given metric for each wavelength in our study population. To answer this question we used
a polychromatic source and then form a weighted average principal component (PC) analysis (Jackson, 1991). This
of the results, analysis indicated that a single PC with the largest charac-
teristic root can account for 65% of the variance between
Metric poly = ∫ V ( λ ) Metric( λ ) d λ (A35) eyes. This first PC is an orthogonal regression line that is a
“line of best fit” since it provides the best account of be-
tween-eye variation of the individual metrics. For our nor-
where the weighting function V(λ) is the luminous effi-
mal, well-corrected eyes the first PC gave approximately
ciency function that describes how visual sensitivity to
equal weighting to all 31 metrics except VOTF (which had
monochromatic light varies with wavelength λ.
about 1/4 the weight of the other metrics). This implies
To the contrary, polychromatic metrics of image quality
that inter-subject variability of metric values (relative to the
for point objects are easily defined by substituting poly-
mean) is about the same for each metric yet each metric
chromatic images for monochromatic images. For example,
emphasizes a different aspect of optical quality. The sign of
the polychromatic luminance point-spread function PSFpoly
the weighting was positive for metrics that increase as opti-
is a weighted sum of the monochromatic spread functions
cal quality increases and negative for metrics that decrease
PSF(x,y,λ),
as optical quality increases. Thus, PC#1 may be interpreted
as an overall metric of optical quality.
PSFpoly = ∫ V ( λ )PSF(x, y, λ ) d λ (A36) One practical use of principal component analysis in
this context is to identify unusual eyes for which the vari-

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 Journal of Vision (2004) 4, 329-351 Thibos, Hong, Bradley & Applegate 349

ous metrics of optical quality do not agree. This outlier Applegate, R. A., Marsack, J. D., Ramos, R., & Sarver, E. J.
analysis is done formally by computing Hotelling’s T- (2003). Interaction between aberrations to improve or
squared statistic for each eye and flagging those values that reduce visual performance. Journal of Cataract and Re-
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15(9), 2545-2551. [PubMed]

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Bracewell, R. N. (1978). The Fourier Transform and Its Appli-
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sion – I: the optical effects of decentering visual tar-
gets or the eye's entrance pupil. In E. Peli (Ed.), Vision
Metric number Models for Target Detection and Recognition (Vol. 2, pp.
313-337). Singapore: World Scientific Press. [Article]
Figure A-10. Correlation matrix. Each square indicates the corre-
lation coefficient for the corresponding pair of metrics. Color bar
Bullimore, M. A., Fusaro, R. E., & Adams, C. W. (1998).
indicates scale of correlation coefficient. Ordering of metrics is The repeatability of automated and clinician refrac-
given in Table 1. tion. Optometry and Vision Science, 75(8), 617-622.
[PubMed]
Burns, S. A., Wu, S., Delori, F., & Elsner, A. E. (1995).
Acknowledgments Direct measurement of human-cone-photoreceptor
alignment. Journal of the Optical Society of America A,
This research was supported by National Institutes of 12(10), 2329-2338. [PubMed]
Health Grant EY-05109 (LNT) and EY 08520 (RAA). We
thank the authors of the Indiana Aberration Study for ac- Campbell, F. W. (1957). The depth of field of the human
cess to their aberration database. We thank. Jacob Rubin- eye. Optica Acta, 4, 157-164.
stein for help with differential geometry issues, Xu Cheng Campbell, F. W., & Green, D. G. (1965). Optical and reti-
for critical comments and help with figure production, and nal factors affecting visual resolution. Journal of Physiol-
Austin Roorda for suggesting the through-focus virtual- ogy, 181, 576-593. [PubMed]
refraction paradigm. Carmo, M. P. (1976). Differential Geometry of Curves and
Surfaces. Englewood Cliffs, NJ: Prentice-Hall.
Commercial relationships: Thibos and Applegate have a
proprietary interest in the development of optical metrics Charman, N., & Jennings, J. A. M. (1976). The optical
predictive of visual performance. quality of the monochromatic retinal image as a func-
Corresponding author: Larry Thibos. tion of focus. British Journal of Physiological Optics, 31,
Email: thibos@indiana.edu. 119-134.
Address: Indiana University, School of Optometry, Bloom- Charman, N., Jennings, J. A. M., & Whitefoot, H. (1978).
ington, IN, USA . The refraction of the eye in relation to spherical aber-
ration and pupil size. British Journal of Physiological Op-
tics, 32, 78-93.
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