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Accuracy and precision of objective refraction from wavefront aberrations
Abstract
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 predicted 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.
History
Received September 28, 2003; published April 23, 2004; corrected April 28, 2005
Citation
Thibos, L. N., Hong, X., Bradley, A., & Applegate, R. A. (2004). Accuracy and precision of objective refraction from wavefront aberrations.
Journal of Vision, 4(4):9, 329-351,
Keywords
visual optics, optical aberrations, refraction, metrics of optical quality
The purpose of a conventional, ophthalmic refraction of
the eye is to determine that combination of spherical and cylindrical lenses
which optimizes visual acuity for distant objects. The underlying assumption of
refraction is that visual acuity is maximized when the quality of the retinal
image is maximized. Furthermore, it is commonly assumed that retinal image
quality is maximized when the image is optimally focused. For these reasons, the
endpoint of a subjective refraction is taken as an operational definition of the
term “best correction” as applied to eyes.
This paper is concerned with the problem of objectively
determining the best correction of an eye from measurements of wavefront
aberrations. Aberrometers measure all of the eye’s monochromatic
aberrations and display the result in the form of an aberration map that
describes the variation in optical path length from source to retinal image
through each point in the pupil. Zernike expansion of an aberration map includes
the second order aberrations of defocus and astigmatism. Thus, one obvious
strategy for objective refraction is to prescribe correcting lenses based on
Zernike coefficients of the second-order.
Unfortunately, the problem is not solved so easily.
Several studies have shown that eliminating the second-order Zernike aberrations
does not necessarily optimize the subjective impression of best-focus nor the
objective measurement of visual performance (Applegate, Ballentine, Gross,
Sarver, & Sarver, 2003; Applegate, Marsack,
Ramos, & Sarver, 2003; Guirao & Williams,
2003; Thibos, Hong, Bradley, & Cheng, 2002). Eliminating second-order Zernike aberrations
is equivalent to minimizing the root mean squared (RMS) wavefront error, but
this minimization does not necessarily optimize the quality of the retinal image
(King, 1968; Mahajan, 1991). Thus a search has begun for alternative
metrics of optical quality that are optimized by subjective refraction when
higher-order aberrations are present.
A variety of problems must be solved when converting an
aberration map into a prescription for corrective lenses or refractive surgery.
One of the most important is a correction for the eye’s chromatic
aberration. Objective aberrometers typically use infrared light, for which the
eye has relatively low refractive power compared to visible
light. Optical models of longitudinal
chromatic aberration (Thibos, Ye, Zhang, & Bradley, 1992) can be extrapolated to estimate the difference
in optical power of the eye between the measurement wavelength and some visible
wavelength, but it is unclear what wavelength should be chosen as a reference
for any given eye. Furthermore, since only one wavelength can be in-focus at a
time, some method is needed to factor in the relative contribution of all
wavelengths, each with a different amount of defocus and a different luminance,
in order to objectively refract an eye for polychromatic
objects.
Another sticky problem is the lack of a
universally-accepted metric of image quality that could be used to establish
objectively the state of optimum-focus for an aberrated eye. One purpose of this
paper is to describe a variety of such metrics based on general principles
described elsewhere (Cheng, Thibos, & Bradley, 2003; Williams, Applegate, & Thibos, 2004). Assuming that consensus agreement could be
achieved for a metric of choice, one still needs to deal with the fact that
identifying the best correction is a multi-dimensional problem in optimization.
Guirao & Williams (Guirao & Williams, 2003) have described an iterative method for finding
the optimum sphere, cylinder and axis parameters that optimize a metric of image
quality. Other possibilities include an objective version of the clinical
technique of refraction by successive elimination. A first approximation would
eliminate the bulk of defocus error by correcting the eye with a spherical lens
of power M, the so-called spherical
equivalent. Next, the eye’s astigmatism is corrected with a cylindrical
lens, followed by a fine-tuning of the spherical lens power if necessary. This
is the basis of most of the methods described below.
A different kind of problem is to incorporate into the
method the refractionist’s rule “maximum plus to best visual
acuity” (Borisch, 1970). According to this
clinical maxim, the spherical refractive error of myopic eyes should be
deliberately under-corrected. The amount of under-correction is not enough to
diminish visual acuity, but it is sufficient to minimize unnecessary
accommodation and to maximize the usable depth of focus (DOF) at distance and
near. These twin goals are achieved by prescribing a spherical lens power that
is slightly less negative (in the case of myopia) or slightly more positive (in
the case of hyperopia) than the lens required to make the retina conjugate to
infinity. Instead, the prescribed lens conjugates the retina with a plane at the
hyperfocal distance, which is the nearest distance the retina can focus on
without significantly reducing visual performance for a target located at
infinity (Campbell, 1957). Consequently, the eye
is left in a slightly myopic state (Figure 1B),
compared to an optimum correction that would place the retina conjugate to
infinity (Figure
1A). Note that the diagram in Figure 1 has been simplified by assuming that any
astigmatism has already been fully corrected using the appropriate cylindrical
lens.
Figure 1. Two criteria for refracting the eye. (A) An optimum
refraction conjugates the retina with infinity. In this case the ideal
correcting lens images infinity at the eye’s far point
(•). (B) A conventional
refraction conjugates the fovea with the eye’s hyperfocal point
(•), which lies closer
to the eye by an amount equal to half the depth-of-field (DOF). In this case the
correcting lens images infinity at a point (o) slightly beyond the eye’s
far point and therefore the eye remains slightly myopic.
Yet another issue is the extent to which neural factors
need to be taken into account when converting an aberration 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 function (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 reported that all five image plane metrics they
considered were more accurate than two pupil-plane metrics in predicting the
optimum subjective refraction for a polychromatic target for a small population
of 6 eyes. Further testing was done on a large population of 146 eyes for which
aberration data for a fixed, 5.7 mm pupil were available in the literature.
Unfortunately, a variety of uncontrolled conditions precluded strong conclusions
from this large population (e.g. possible fluctuations of accommodation, unknown
pupil size during subjective refraction, binocular refractions that likely
yielded sub-optimal acuity endpoints) but nevertheless the authors found a close
correlation between subjective and objective refractions computed from
image-plane metrics. Although visual performance during refraction presumably
depended on some combination of optical and neural factors, they found that
optimizing the optical image without considering neural factors led to accurate
prediction of the outcome of subjective refraction. However, no assessment of
the precision of these predictions was reported.
The purpose of our study was to evaluate two general
approaches to converting an aberration map into a conventional
sphero-cylindrical prescription. The first approach is a surface-fitting
procedure designed to find the nearest sphero-cylindrical approximation to the
actual wavefront aberration map. The second approach involves a virtual
through-focus experiment in which the computer adds or subtracts various amounts
of spherical or cylindrical wavefronts to the aberration map until the optical
quality of the eye is maximized. Preliminary accounts of this work have been
presented (Thibos, Bradley, & Applegate, 2002; Thibos, Hong, & Bradley, 2001).
We define the
equivalent
quadratic of a wavefront aberration map
as that quadratic (i.e. a sphero-cylindrical) surface which best represents the
map. This idea of approximating an arbitrary surface with an equivalent
quadratic is a simple extension of the common ophthalmic technique of
approximating a sphero-cylindrical surface with an equivalent sphere. Two
methods for determining the equivalent quadratic from an aberration map are
presented
next.
One common way to fit an arbitrarily aberrated
wavefront with a quadratic surface is to minimize the sum of squared deviations
between the two surfaces. This least-squares fitting method is the basis for
Zernike expansion of wavefronts. Because the Zernike expansion employs an
orthogonal set of basis functions, the least-squares solution is given by the
second-order Zernike coefficients, regardless of the values of the other
coefficients. These second-order Zernike coefficients can be converted to a
sphero-cylindrical prescription in power vector notation using Equation
1,
where cnm is the
nth order Zernike
coefficient of meridional frequency m,
and r is pupil radius. The power vector
notation is a cross-cylinder convention that is easily transposed into
conventional minus-cylinder or plus-cylinder formats used by clinicians (see
eqns 22, 23 of (Thibos, Wheeler, & Horner, 1997).
Curvature is the property of wavefronts that determines
how they focus. Thus, another reasonable way to fit an arbitrary wavefront with
a quadratic surface is to match the curvature of the two surfaces at some
reference point. A variety of reference points could be selected, but the
natural choice is the pupil center. Two surfaces that are tangent at a point and
have exactly the same curvature in every meridian 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 the osculating quadratic from the
Zernike coefficients of the wavefront (Thibos et al., 2002). This solution is obtained by computing the
curvature at the origin of the Zernike expansion of the Seidel formulae for
defocus and astigmatism. This process effectively collects all r2
terms from the various Zernike modes. We used the OSA definitions of the Zernike
polynomials, each of which has unit variance over the unit circle (Thibos,
Applegate, Schwiegerling, & Webb, 2000). The
results given in Equation 2 are truncated at the sixth Zernike
order but could be extended to higher orders if
warranted.
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 procedure is easily simulated mathematically by adding a
spherical wavefront to the eye’s aberration map and then computing the
retinal image using standard methods of Fourier optics as illustrated in Movie 1. The curvature of the added wavefront can be
systematically varied to simulate a through-focus experiment that varies the
optical quality of the eye+lens system over a range from good to bad. Given a
suitable metric of optical quality, this computational procedure yields the
optimum power M of that spherical
correcting lens needed to maximize optical quality of the corrected eye. With
this virtual spherical lens in place, the process can be repeated for
through-astigmatism calculations to determine the optimum values of
J0 and
J45 needed to maximize image
quality. If necessary, a second iteration could be used to fine-tune results by
repeating the above process with these virtual lenses in place. However, the
analysis reported below did not include a second iteration.
Movie 1.
Dynamic simulation of the through-focus method of objective refraction. To
determine the optimum value M of a
spherical defocusing lens, a pre-determined sequence of M-values are used to
modulate the wavefront map in the same way that a real lens alters the
eye’s wavefront aberration function. From the new aberration map we
compute the retinal point-spread function (PSF), optical transfer function
(OTF), and retinal image of an eye chart. Scalar metrics of optical quality are
used to optimize focus (M). The process is then repeated to optimize astigmatism
parameters J0,
J45. This example is for an
eye with 0.1 mm of spherical aberration.
The computational method described above captures the
essence of clinical refraction by mathematically simulating the effects of
sphero-cylindrical lenses of various powers. Our method is somewhat simpler to
implement than that described by Guirao & Williams (Guirao & Williams,
2003) who used an iterative searching method to
determine that combination of spherical and cylindrical lenses which maximizes
the eye’s optical quality. Regardless of which searching algorithm is
used, a suitable metric of optical quality is required as a merit function.
Guirao and Williams used 5 such metrics of image quality. In Appendix A we expand their list to 31 metrics
by systematically pursuing three general approaches to quantifying optical
quality: (1) wavefront quality, (2) retinal image quality for point objects, and
(3) retinal image quality for grating objects. Implementation of image sharpness
metrics for extended objects, such as a letter chart, (Fienup & Miller, 2003; Hultgren, 1990)
have been left for future work. Several of the implemented metrics include a
neural component that takes into account the spatial filtering of the retinal
image imposed by the observer’s visual system. Strictly speaking, such
metrics should be referred to as metrics of neuro-optical quality or visual
quality, but for simplicity we use the term “optical quality metric”
generically. For each of these 31 metrics we used the virtual refraction
procedure described above to determine (to the nearest 1/8 D) the values of
M,
J0 and
J45 required to maximize the
metric. These objective refractions were then compared with conventional
subjective refractions. A listing of acronyms for the various refraction
methods is given in Table
1.
To judge the success of an objective method of
refraction requires a gold standard for comparison. The most clinically relevant
choice is a subjective refraction performed 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 aberration
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 Aberration Study is given next.
Subjective refractions were performed to the nearest
0.25D on 200 normal, healthy eyes from 100 subjects using the standard
optometric protocol of maximum plus to best visual acuity. Accommodation was
paralyzed with 1 drop of 0.5% cyclopentalate during the refraction. Optical
calculations were performed for the fully dilated pupil, which varied between
6-9 mm for different eyes. The refractive correction was taken to be that
sphero-cylindrical lens combination which optimally corrected astigmatism and
conjugated the retina with the eye’s hyperfocal point (Figure 1b). This prescribed refraction was then
implemented with trial lenses and worn by the subject during subsequent
aberrometry (l=633 nm). This experimental design emphasized the effects
of higher-order aberrations by minimizing the presence of uncorrected
second-order aberrations. The eye’s longitudinal chromatic aberration was
taken into account by the different working distances used for aberrometry and
subjective refraction as illustrated in Figure 2. Assuming the eye was well
focused for 570 nm when viewing the polychromatic eye chart at 4 m, the eye
would also have been focused at infinity for the 633nm laser light used for
aberrometry (Thibos et al., 1992).
Figure 2. Schematic diagram of optical condition of the Indiana
Aberration Study. Yellow light with 570 nm wavelength is assumed 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
during aberrometry, the predicted refraction was
M =
J0 =
J45 = 0. The level of
success achieved by the 33 methods of objective refraction described above was
judged on the basis of precision and accuracy at matching these predictions (Figure 3). Accuracy for the spherical component of
refraction was computed as the population mean of
M as determined from objective
refractions. Accuracy for the astigmatic component of refraction was computed as
the population mean of (Bullimore, Fusaro, & Adams, 1998) vectors. Precision is a measure of the
variability in results and is defined for
M as twice the standard deviation of
the population values, which corresponds to the 95% limits of agreement (LOA)
(Bland & Altman, 1986). The confidence region
for astigmatism is an ellipse computed for the bivariate distribution of
J0 and
J45. This suggests a
definition of precision as the geometric mean of the major and minor axes of the
95% confidence ellipse.
Figure 3. Graphical
depiction of the concepts of precision and accuracy as applied to the
1-dimensional problem of estimating spherical power (left column of diagrams)
and the 2-dimensional problem of estimating astigmatism (right column of
diagrams).
In our view, accuracy and precision are equally
important for refraction. A method that is precise but not accurate will yield
the same wrong answer every time. Conversely, a method that is accurate but not
precise gives different answers every time and is correct only on average. Thus
we seek a method that is both accurate and precise. However, one might argue
that lack of accuracy implies a systematic bias that could be removed by a
suitable correction factor applied to any individual eye. One way to obtain such
a correction factor is to examine the population statistics of a large number of
eyes, as we have done in this study. Any systematic bias obtained for this group
could then be used as a correction factor for future refractions, assuming of
course that the individual in question is well represented by the population
used to determine the correction factor. Although this may be an expedient
solution to the problem of objective refraction, it lacks the power of a
theoretically sound account of the reasons for systematic biases in the various
metrics of optical
quality.
The two methods for determining the equivalent
quadratic surface for a wavefront aberration map gave consistently different
results. A frequency histogram of results for the least-squares method (Figure 4A) indicated an average spherical refractive
error of M = –0.39 D. In other words, this objective method predicted
the eyes were, on average, significantly myopic compared to subjective
refraction. To the contrary, the method based on paraxial curvature matching (Figure 4B) predicted an average refractive error
close to zero for our population. Both methods accurately predicted the expected
astigmatic refraction as shown by the scatter plots and 95% confidence ellipses
in Figure
5.
Figure 4. Frequency
distribution of results for the least-squares method for fitting the wavefront
aberration map with a quadratic surface. Dilated pupil size ranged from 6 to 9
mm across the population.
Figure 5. Scatter plots of
(A) the least-squares fit of the wavefront over the entire pupil and (B)
paraxial curvature matching methods of determining the two components of
astigmatism. Circles show the results for individual eyes, green cross indicates
the mean of the 2-dimensional distribution, and ellipses are 95% confidence
intervals. Precision is the geometrical mean of the major and minor axes of the
ellipse.
Computer simulation of through-focus experiments to
determine that lens (either spherical or astigmatic) which optimizes image
quality are computationally intensive, producing many intermediate results of
interest but too voluminous to present here. One example of the type of
intermediate results obtained when optimizing the pupil fraction metric PFWc
(see Table 1 for a list of acronyms) is shown in Figure 6A. For each lens power over the range
–1 to +1 D (in 0.125 D steps) a curve is generated relating RMS wavefront
error to pupil radius. Each of these curves crosses the criterion level
(l/4 in our calculations) at some radius value. That radius is
interpreted as the critical radius since it is the largest radius for which the
eye’s optical quality is reasonably good. The set of critical radius
values can then be plotted as a function of defocus, as shown in Figure 6B. This through-focus function peaks at some
value of defocus, which is taken as the optimum lens for this eye using this
metric. In this way the full dataset of Figure 6
is reduced to a single number.
Table 1. Listing of acronyms for
refraction methods. Ordering is that used in correlation matrices (Figures 8, A8).
Figure 6. Rank ordering
(based on accuracy) of 33 methods for predicting spherical refractive error. Red
symbols indicate means for metrics based on wavefront quality. Black symbols
indicate mean for metrics based on image quality. Error bars indicate ± 1
standard deviation of the population. Numerical data are given in Table 2.
Similar calculations were then repeated for other eyes
in the population to yield 200 estimates of the refractive error using this
particular metric. A frequency histogram of these 200 values similar to those in
Figure 4 was produced for inspection by the
experimenters. Such histograms were then summarized by a mean value, which we
took to be a measure of accuracy, and a standard deviation, which (when doubled)
was taken as a measure of
precision.
The accuracy and precision of the 31 methods for
objective refraction based on optimizing metrics of optical quality, plus the
two methods based on wavefront fitting, are displayed in rank order in Figure 7. Mean accuracy varied from –0.50 D to
+0.25 D. The 14 most accurate methods 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).
Figure 7. An example of
intermediate results for the through-focus calculations needed to optimize the
pupil fraction metric PFWc. (A) The RMS value is computed as a function of pupil
radius for a series of defocus values added to the wavefront aberration function
of this eye. The pupil size at the intersection points of each curve with the
criterion level of RMS are plotted as a function of lens power in (B). The
optimum correcting lens for this eye is the added spherical power that maximized
the critical pupil diameter (and therefore maximized PFWc) which in this example
is +0.125 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 confidence range given by the mean ± 0.5 D. The most
precise method was PFSc (±0.49D), which was statistically significantly
better than the others (F-test for
equality of variance, 5% significance level). Precision of the next 14 methods
in rank ranged from ±0.58D to ±0.65D. These values were statistically
indistinguishable from each other. This list of the 15 most precise methods
included several examples from each of the three categories of wavefront
quality, point-image quality, and grating-image quality. Rank ordering of all
methods for predicting defocus is given in Table
2.
Table 2. Rank ordering of methods
for predicting spherical equivalent M
based on accuracy and precision. Acronyms in red type are wavefront quality
methods. Brief descriptions of acronyms are given in Table 1.
Detailed descriptions are in Appendix. Units are diopters.
A similar process was used to determine the accuracy
for estimating astigmatism. We found that all methods except one (PFCc) had a
mean error across the population of less than 1/8 D. This accuracy is the best
we could reasonably expect, given that the subjective refractions and the
virtual refractions used to predict subjective refractions were both quantized
at 1/8 D of cross-cylinder power. Precision of astigmatism predictions was
typically better than precision for predicting defocus. The precision of all
metrics for predicting astigmatism ranged from ±0.32D to ±1.0D and the
15 best methods were better than ±0.5D. Rank ordering of all methods for
predicting astigmatism is given in Table
3.
Table 3. Rank ordering of methods
for predicting astigmatism parameters J0 and J45 jointly. Acronyms in red type
are wavefront quality methods. Brief descriptions of acronyms are given in Table 1. Detailed descriptions are in Appendix. Units are
diopters.
In comparing the precision for predicting defocus and
astigmatism we found that 7 metrics were in the top-15 list for both types of
prediction. Five of these were also accurate to within 1/8 D for predicting both
defocus and astigmatism. Thus 5 metrics (PFSc, PFWc, VSMTF, NS, and PFCt)
emerged as reasonably accurate and among the most precise. Three of these
successful metrics were pupil plane metrics and two were image plane metrics.
These results demonstrate that accurate predictions of subjective refractions are possible with pupil plane metrics.
However, such metrics do not include the process of image formation that occurs
in the eye, a process that must influence subjective image quality. For this
reason, image-plane metrics of visual quality are more germane to vision models
of the refraction process that seek to capture the subjective notion of a
well-focused retinal image (Williams, Applegate, & Thibos, 2004).
One implication of the results presented above is that
different methods of objective refraction that yield similar refractions on
average are likely to be statistically correlated. We tested this prediction by
computing the correlation coefficient between all possible pairs of methods for
predicting M. The resulting correlation
matrix is visualized in Figure 8. For example, the left-most
column of tiles in the matrix represents the Pearson correlation coefficient
r between the first
objective refraction method in the list (RMSw) and all other methods in the
order specified in Table 1. Notice that the values of
M predicted by optimizing RMSw are
highly correlated with the values returned by methods 3 (RMSs), 8 (Bave), 19
(ENT), and 32 (least-squares fit). As predicted, all of these metrics are
grouped at the bottom of the ranking in Figure 7.
To the contrary, refractions using RMSw are poorly correlated with values
returned by methods 4 (PFWc), 9 (PFCt), 21 (VSX), 24 (SFcOTF), and 33 (Curvature
fit). All of these metrics are grouped at the top of the ranking in Figure 7, which further supports this connection
between accuracy and correlation. A similar analysis of the correlation matrix
for astigmatism parameters is not as informative because there was very little
difference between the various methods for predicting
J0 and
J45.
 Figure 8.
Correlation matrix for values of M
determined by objective refraction. Metric number is given in Table 1.
Another interesting feature of Figure
8 is that some refraction methods (e.g. PFCc, VOTF, VNOTF) are very poorly
correlated with all other methods. This result for metric PFCc is explained by
the fact that PFCc was the only metric to produce hyperopic refractions in the
vicinity of M=+0.25D. However, this argument does not apply to the other two
examples that are poorly correlated with most other metrics even though these
other metrics produced similar refractions on average (e.g. 20 (NS), 7 (PFSc),
and 23 (AreaMTF)). This result suggests that maximizing metrics VOTF and VNOTF
optimizes a unique aspect of optical and visual quality that is missed by other
metrics. In fact, these two metrics were specifically designed to capture
infidelity of spatial phase in the retinal
image.
The least-squares method for fitting an aberrated
wavefront with a spherical wavefront is the basis of Zernike expansion to
determine the defocus coefficient. The failure of this method to accurately
predict the results of subjective refraction implies that the Zernike
coefficient for defocus is an inaccurate indicator of the spherical equivalent
of refractive error determined by conventional subjective refractions. On
average, this metric predicted that eyes in our study were myopic by -0.39D
when in fact they were well corrected.
To the contrary, matching paraxial curvature accurately
predicted the results of subjective refraction. This method is closely related
to the Seidel expansion of wavefronts because it isolates the purely parabolic
(r2) term. It also
corresponds 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 predicting 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 defocus. 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 factors that could potentially make all of
the predictions of defocus equally accurate. While this might seem a reasonable
approach to improving accuracy, it may prove cumbersome 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 wavelength of light was well focused on the retina during subjective
refraction with a polychromatic stimulus. We chose 570 nm as our reference
wavelength based on theoretical and experimental evidence (Charman & Tucker,
1978; Thibos & Bradley, 1999) but the actual value is unknown. Changing this
reference wavelength by just 20 nm to 550 nm would cause a 0.1 D shift in
defocus, which is a significant fraction of the differences in accuracy between
the various metrics.
A second source of bias may be attributed to the
difference between optimal and conventional refraction methods. The objective
refraction procedures described in this paper are designed to determine the
optimum refraction (Figure 1a) whereas the
subjective refractions were conventional (Figure
1b). The difference between the two endpoints is half the depth-of-focus
(DOF) of the eye. The DOF for subjects in the Indiana Aberration Study is
unknown, but we would anticipate a value of perhaps ±0.25D (Atchison,
Charman, & Woods, 1997) which is about half
the total range of focus values spanned in Figure
7. Accordingly, we may account for the results in Figure 7 by supposing that the curvature matching
technique happens to locate the far end of the DOF interval (which is located at
optical infinity in a conventional refraction) whereas some middle-ranking
metric (such as VSOTF) locates the middle of the DOF, located at the hyperfocal
distance. This inference is consistent with the fact that most eyes in the
Indiana Aberration Study had positive spherical aberration. Such eyes have less
optical power for paraxial rays than for marginal rays. Consequently, the retina
will appear to be conjugate to a point that is beyond the hyperfocal point if
the analysis is confined to the paraxial rays.
The preceding arguments suggest that the superior
accuracy of the curvature method for determining the spherical equivalent of a
conventional refraction is due to a bias in this method that favors the far end
of the eye’s DOF. In short, curvature
matching (and several other metrics with similar accuracy) is a biased method
that successfully predicts a biased endpoint. By the same argument, the
biased curvature method is not expected to predict astigmatism accurately
because conventional refractions are unbiased for astigmatism. Although this
line of reasoning explains why the paraxial curvature method will locate a point
beyond the hyperfocal point, we lack a convincing argument for why the located
point should lie specifically at infinity. Perhaps future experiments that
include measurement of the DOF as well as the hyperfocal distance will clarify
this issue and at the same time help identify objective methods for determining
the hyperfocal distance.
Pursuing the above line of reasoning suggests that some
metric near the bottom of the accuracy ranking, such as RMSw, locates the near
end of the DOF. This accounting is consistent with the findings of Guirao and
Williams (Guirao & Williams, 2003) and of
Cheng et al. (Cheng, Bradley, &
Thibos, 2004) that the optimum focus lies
somewhere between the more distant paraxial focus and the nearer RMS focus.
Taken together, the least-squares and curvature fitting methods would appear to
locate the two ends of the DOF interval. While perhaps a mere coincidence, if
this intriguing result could be substantiated theoretically then it might become
a useful method to compute the DOF from the wavefront aberration map for
individual eyes.
A variety of other factors may also contribute to the
range of inaccuracies documented in Figure 7. For
example, all of the image quality metrics reported in this paper are based on
monochromatic light. Generalizing these metrics to polychromatic light might
improve the predictions of the subjective refraction. Inclusion of
Stiles-Crawford apodization in the calculations might also improve the
predictions. Also, it may be unrealistic to think that a single metric will
adequately capture the multi-faceted notion of best-focus. A multi-variate
combination of metrics which captures different aspects of optical image quality
may yield better predictions (Williams et al., 2004). Those metrics that included a neural
component were configured with the same neural filter, when in fact different
individuals are likely to have different neural filters. Furthermore, the
characteristics of the neural filter are likely to depend on stimulus
conditions. Koomen et al. (Koomen,
Scolnik, & Tousey, 1951) and Charman
et al. (Charman, Jennings, &
Whitefoot, 1978) found that pupil size affects
subjective refraction differently under photopic and scotopic illumination. They
suggested that this might be due to different neural filters operating at
photopic and scotopic light levels. A change in neural bandwidth of these
filters would alter the relative weighting given to low and high spatial
frequency components of the retinal image, thereby altering the optimum
refraction. This idea suggests future ways to test the relative importance of
the neural component of metrics of visual quality described here.
Variability in the gold standard of subjective
refraction is another likely source of disagreement between objective and
subjective refractions. The range of standard deviations for predicting
M across all metrics was only 1/8 D
(0.29-0.42 D), indicating that the precision of all metrics was much the same.
This suggests that the precision of objective refraction might be dominated by a
single, underlying source of variability. That source might in fact be
variability in the subjective refraction. Bullimore
et al found that the 95% limit of
agreement for repeatability of refraction is ± 0.75D, which corresponds to
a standard deviation of 0.375 D (Bullimore et al., 1998). If the same level of variability were present
in our subjective refractions, then uncertainty in determining the best
subjective correction would have been the dominant source of error. It is
possible, therefore, that all of our objective predictions are extremely precise
but this precision is masked by imprecision of the gold standard of subjective
refraction. If so, then an objective wavefront analysis that accurately
determines the hyperfocal point and the DOF with reduced variability could
become the new gold standard of refraction.
The metrics of image quality described in this paper
have a potential utility beyond objective refraction. For example, Cheng
et al. (Cheng et al., 2004) and Marsack
et al. (Applegate, Marsack, &
Thibos, 2004) both used the same implementation of
these metrics described below (see Appendix) to predict the change in visual
acuity produced when selected, higher-order aberrations are introduced into an
eye. The experimental design of the Cheng study was somewhat simpler in that
monochromatic aberrations were used to predict monochromatic visual performance,
whereas Marsack used monochromatic aberrations to predict polychromatic
performance. Nevertheless, both studies concluded that changes in visual acuity
are accurately predicted by the pupil plane metric PFSt and by the image plane
metric VSOTF. Furthermore, both studies concluded that three of the least
accurate predictors were RMSw, HWHH, and VOTF. In addition, the Cheng study
demonstrated that, as expected, those metrics which accurately predicted changes
in visual acuity also predicted the lens power which maximized acuity in a
through-focus experiment. This was an important result because it established a
tight link between variations in monochromatic acuity and monochromatic
refraction.
The superior performance of metric VSOTF is also
consistent with the present study. This metric lies in the middle of the
accuracy ranking for predicting M in a
conventional refraction, which suggests that it would have accurately predicted
M in an optimum refraction. (This point
is illustrated graphically in Figure 5 of the Cheng
et al. paper.) Furthermore, present
results show that VSOTF is one of the most precise methods for estimating
M, which suggests 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 images 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 associated with these
metrics would have contributed to the poor performance recorded in those
companion
studies.
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 introduction of
controlled amounts of selected, higher-order aberrations into polychromatic
letters (Applegate et al., 2004).
A perfect optical system has a flat wavefront
aberration map and therefore metrics of wavefront quality are designed to
capture the idea of flatness. An aberration map is flat if its value is
constant, or if its slope or curvature is zero across the entire pupil. Since a
wavefront, its slope, and its curvature each admits to a different optical
interpretation, we sought meaningful scalar metrics based on all three: the
wavefront aberration map, the slope map, and the curvature map. Programs for
computing the metrics were written in Matlab (The Mathworks, Inc.) and tested
against known
examples.
Wavefront
error describes optical path differences across the pupil that give rise
to phase errors for light entering the eye through different parts of the pupil.
These phase errors produce interference effects that degrade the quality of the
retinal image. An example of a wave aberration map is shown in Figure A-1. Two common metrics of wavefront flatness
follow.
Figure A-1. A
theoretical wavefront aberration map for 1mm RMS of the third-order
aberration coma over a 6mm pupil.
where w(x,y) is the wavefront aberration
function defined over pupil coordinates x,y,
A = pupil area, and the integration is
performed over the domain of the entire pupil. Computationally, RMSw is just the
standard deviation of the values of wavefront error specified at various pupil
locations.
PV is the difference between the highest and
lowest points in the aberration
map.
Wavefront
slope is a vector-valued function of pupil position that requires two
maps for display, as illustrated in Figure
A-2. One map shows the slope in the
horizontal (x) direction and the other map shows the slope in the vertical (y)
direction. (Alternatively, a polar-coordinate scheme would show the radial slope
and tangential slope.) Wavefront slopes may be interpreted as transverse ray
aberrations that blur the image. These ray aberrations can be conveniently
displayed as a vector field (lower right diagram). The base of each arrow in
this plot marks the pupil location and the horizontal and vertical components of
the arrow are proportional to the partial derivatives of the wavefront map. If
the field of arrows is collapsed so that all the tails superimpose, the tips of
the arrows represent a spot diagram (lower right diagram) that approximates the
system point-spread function (PSF).
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).
The root-mean-squared value of a slope map is a measure
of the spreading of light rays that blur the image in one direction. The total
RMS value computed for both slope maps taken together is thus a convenient
metric of wavefront quality that may be interpreted in terms of the size of the
spot
diagram.
where wx=dw/dx and
wy=dw/dy are the partial spatial derivatives (i.e. slopes) of w(x,y)
and A = pupil area.
Wavefront
curvature describes focusing errors that blur the image. To form a good
image at some finite distance, wavefront curvature must be the same everywhere
across the pupil. A perfectly flat wavefront will have zero curvature
everywhere, which corresponds to the formation of a perfect image at infinity.
Like wavefront slope, wavefront curvature is a vector-valued function of
position that requires more than one map for display (Figure A-3). Curvature varies not only with pupil
position but also with orientation at any given point on the
wavefront. Figure A-3. Curvature maps derived from the
wavefront in Figure 1. Calibration bars have units
of diopters.
Fortunately, Euler’s classic formula of
differential geometry assures us that the curvature in any meridian can be
inferred from the principal curvatures (i.e. curvatures in the orthogonal
meridians of maximum and minimum curvature) at the point in question (Carmo, 1976). The principal curvatures at every point can
be derived from maps of mean curvature M(x,y) and Gaussian curvature G(x,y) as
follows.
where the principal curvature maps
k1(x,y), k2(x,y) are computed from
M and
G
using
The Gaussian and mean curvature maps may be
obtained from the spatial derivatives of the wavefront aberration map using
textbook formulas (Carmo, 1976). Given the
principal curvature maps, we can reduce the dimensionality of wavefront
curvature by computing blur strength at every pupil location. The idea of blur
strength is to think of the wavefront locally as a small piece of a quadratic
surface for which a power vector representation can be computed (Thibos et al.,
1997). A power vector P (Bullimore et al., 1998) is a 3-dimensional vector whose coordinates
correspond to the spherical equivalent (M), the normal component of astigmatism
(J0) and the oblique component of astigmatism (J45).
Experiments have shown that the length of the power vector, which is the
definition of blur strength, is a good scalar measure of the visual impact of
sphero-cylindrical blur (Raasch, 1995). Thus, a
map of the length of the power-vector representation of a wavefront at each
point in the pupil may be called a blur-strength map (Figure A-3).
To compute the blur-strength map we first use the
principal curvature maps to compute the astigmatism map
and then combine the astigmatism map with the
mean curvature map using the Pythagorean formula to produce a blur strength map
The spatial average of this blur strength map
is a scalar value that represents the average amount of focusing error in the
system that is responsible for image
degradation,
In addition to the 4 metrics described above, another 6
metrics of wavefront quality can be defined based on the concept of
pupil fraction. Pupil fraction is
defined as the fraction of the pupil area for which the optical quality of the
eye is reasonably good (but not necessarily diffraction-limited). A large pupil
fraction is desirable because it means that most of the light entering the eye
will contribute to a good-quality retinal
image.
Two general methods for determining the area of the
good pupil are illustrated in Figure A-4. The
first method, called the critical pupil or central pupil method, examines the
wavefront inside a sub-aperture that is concentric with the eye’s pupil
(Corbin, Klein, & van de Pol, 1999; Howland
& Howland, 1977). We imagine starting with a
small sub-aperture where image quality is guaranteed to be good (i.e.
diffraction-limited) and then expanding the aperture until some criterion of
wavefront quality is reached. This endpoint is the
critical diameter, which can be used to
compute the pupil fraction (critical pupil method) as
follows
Figure A-4. Pupil
fraction method for specifying wavefront quality. Red circle in the left diagram
indicates the largest concentric sub-aperture for which the wavefront has
reasonably good quality. 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. l/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 represented by the blur strength map,
PFCc = PFc when critical pupil is defined as the concentric area for which Bave < criterion (e.g. 0.25D)
The second general method for determining the area of
the good pupil is called the tessellation or whole pupil method. We imagine
tessellating the entire pupil with small sub-apertures (about 1% of pupil
diameter) and then labeling each sub-aperture as good or bad according to some
criterion (Figure A-4, right-hand diagram). The
total area of all those sub-apertures labeled good defines the area of the good
pupil from which we compute pupil fraction
as
As with the concentric metrics, implementation of Equation 11 requires criteria for deciding if the wavefront
over a sub-aperture is good. For example, the criterion could be based on the
wavefront aberration function,
Alternatively, the criterion could be based on
wavefront slope,
PFSt = PFt when a good sub-aperture satisfies the criterion 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,
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 magnitude of
the inverse Fourier transform of the pupil function P(x,y), defined
as
where
k is the wave number
(2p/wavelength) and A(x,y) is 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 function is usually
omitted. However, when computing the visual effectiveness of the retinal image,
the waveguide nature of cones must be taken into account. These waveguide
properties cause the cones to be more sensitive to light entering the middle of
the pupil than to light entering at the margin of the pupil (Burns, Wu, Delori,
& Elsner, 1995; Roorda & Williams, 2002; Stiles & Crawford, 1933). It is common practice to model this
phenomenon as an apodizing filter with transmission A(x,y) in the pupil plane
(Atchison, Joblin, & Smith, 1998; Bradley
& Thibos, 1995; Zhang, Ye, Bradley, &
Thibos, 1999).
Figure A-5. Measures of image quality for point
objects are based on contrast and compactness of the image.
Scalar metrics of image quality that measure the
quality of the PSF in aberrated eyes are designed to capture the 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
of good quality. The last 6 metrics measure contrast and in every case large
values of the metric indicate a high-contrast PSF of good quality. Most of the
metrics are completely optical in character, but a few also include knowledge of
the neural component of the visual system. Several of these metrics are
2-dimensional extensions of textbook metrics defined for 1-dimensional impulse
response functions (Bracewell, 1978). Many of the
metrics are normalized by diffraction-limited values and therefore are
unitless.
D50 = diameter of a circular area centered on PSF peak which captures 50% of the light energy (arcmin)
The value of D50 is equal to the radius
r, where
r is defined implicitly by:
where PSFN is the normalized (i.e.
total intensity = 1) point-spread function with its peak value located at r = 0.
This metric ignores the light outside the central 50% region, and thus is
insensitive to the shape of the PSF
tails.
The equivalent width of the PSF is the diameter of the
circular base of that right cylinder which has the same volume as the PSF and
the same height. The value of EW is given by:
where
 are the coordinates of
the peak of the PSF. In this and following equations, x,y are spatial
coordinates of the retinal image, typically specified as visual angles subtended
at the eye’s nodal point. Note that although EW describes spatial
compactness, it is computed from PSF contrast. As the height falls the width
must increase to maintain a constant volume under the
PSF.This metric is analogous to the moment of inertia of a
distribution of mass. It is computed
as
Unlike D50 above, this compactness metric is
sensitive to the shape of the PSF tails. Large values of SM indicate a rapid
roll-off of the optical transfer function at low spatial frequencies (Bracewell,
1978).
This metric is the average width of every cross-section
of the PSF. It is computed
as
where C(x,y) = 1 if PSF(x,y) > max(PSF)/2,
otherwise C(x,y) = 0. A 1-dimensional version of this metric has been used on
line spread functions of the eye (Charman & Jennings, 1976; Westheimer & Campbell, 1962).
This metric is the HWHH of the autocorrelation of the
PSF. It is computed
as
where Q(x,y) = 1 if
> max()/2, otherwise Q(x,y) = 0. In this expression,
is the autocorrelation of the PSF.
This widely-used metric is typically defined with
respect to the peak of the PSF, rather than the coordinate origin. It is
computed
as
where
 is the
diffraction-limited PSF for the same pupil diameter.
The value of this metric is the percentage of total
energy falling in an area defined by the core of a diffraction-limited
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 diameter. An
alternative domain of interest is the entrance aperture of cone photoreceptors.
Similar metrics have been used in the study of depth-of-focus (Marcos, Moreno,
& Navarro, 1999).
This metric measures the variability of intensities at
various points in the PSF,
where PSFDL is the
diffraction-limited point-spread function. The domain of integration is a
circular area centered on the PSF peak and large enough in diameter to capture
most of the light in the
PSF.
This metric is inspired by an information-theory
approach to optics (Guirao & Williams, 2003).
This metric was introduced by Williams as a way to
capture the effectiveness of a PSF for stimulating the neural portion of the
visual system (Williams, 2003). This is achieved
by weighting the PSF with a spatial sensitivity function that represents the
neural visual system. The product is then integrated over the domain of the PSF.
Here we normalize the result by the corresponding value for a
diffraction-limited PSF to achieve a metric that is analogous to the Strehl
ratio computed for a neurally-weighted PSF,
where g(x,y) is a bivariate-Gaussian, neural
weighting-function. A profile of this weighting function (Figure A-6)
shows that it effectively ignores light outside of the central 4 arc
minutes of the PSF. Figure A-6. Neural
weighting functions used by NS and VSX.
Like the neural sharpness metric, the visual Strehl
ratio is an inner product of the PSF with a neural weighting function normalized
to the diffraction-limited case. The difference between NS and VSX is in the
choice of weighting functions (Figure A-6).
where N(x,y) is a bivariate neural weighting
function equal to the inverse Fourier transform of the neural contrast
sensitivity function for interference fringes (Campbell & Green, 1965). With this metric, light outside of the
central 3 arc minutes of the PSF doubly detracts from image quality because it
falls outside the central core and within an inhibitory surround. This is
especially so for light just outside of the central 3 arc minutes in that
slightly aberrated rays falling 2 arc minutes away from the PSF center are more
detrimental to image quality than highly aberrated rays falling farther from the
center.
Unlike point objects, which can produce an infinite
variety of PSF images depending on the nature of the eye’s aberrations,
small patches of grating objects always produce sinusoidal images no matter how
aberrated the eye. Consequently, there are only two ways that aberrations can
affect the image of a grating patch: they can reduce the contrast or translate
the image sideways to produce a phase-shift, as illustrated in Figure A-7. In general, the amount of contrast
attenuation and the amount of phase shift both depend on the gratings spatial
frequency. This variation of image contrast with spatial frequency for an object
with 100% contrast is called a modulation transfer function (MTF). The variation
of image phase shift with spatial frequency is called a phase transfer function
(PTF). Together, the MTF and PTF comprise the eye’s optical transfer
function (OTF). The OTF is computed as the Fourier transform of the PSF.
Optical theory tells us that any object can be
conceived as the sum of gratings of various spatial frequencies, contrasts,
phases and orientations. In this context we think of the optical system of the
eye as a filter that lowers the contrast and changes the relative position of
each grating in the object spectrum as it forms a degraded retinal image. A
high-quality OTF is therefore indicated by high MTF values and low PTF values.
Scalar metrics of image quality in the frequency domain are based on these two
attributes of the OTF.
Figure A-7.
Measures of image quality for grating objects are based on contrast and phase
shifts in the image. Upper row depicts a high-quality image of a grating object.
Lower row depicts a low quality image with reduced contrast and a 180 deg phase
shift. Left column shows the grating images and right column show horizontal
traces of intensity through the corresponding images. Red lines are reference
marks that highlight the phase shifts that can occur in blurred images.SFcMTF =
spatial frequency cutoff of radially-averaged modulation-transfer function
(rMTF)
Cutoff SF is defined here as the intersection of the
radially averaged MTF (rMTF) and the neural contrast threshold function (Thibos,
1987). The rMTF is computed by integrating the
full 2-dimensional MTF over orientation. This metric does not capture spatial
phase errors in the image because rMTF is not affected by the PTF portion of the
OTF.
where
and OTF(f,f) is the optical transfer
function for spatial frequency coordinates
f (frequency) and f
(orientation). A graphical depiction of SFcMTF is shown
in Figure
A-8.
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 computing rOTF, this metric is intended to
capture spatial phase errors in the
image.
where
Figure A-8. Radial
MTF for a defocused optical system, showing intersection with neural threshold
function that defines cutoff spatial frequency metric SFcMTF. Shaded area below
the MTF and above the neural threshold is the area of visibility specified in
the definition of metric AreaMTF.
and OTF(f,f) is the optical transfer
function for spatial frequency coordinates
f (frequency) and f
(orientation). Since the OTF is a complex-valued function, integration is
performed separately for the real and imaginary components. Conjugate symmetry
of the OTF ensures that the imaginary component vanishes, leaving a real-valued
result. A graphical depiction of SFcOTF is shown
in Figure
A-9. Figure A-9. Radial
OTF for a defocused optical system, showing intersection with neural threshold
function to define cutoff spatial frequency metric SFcOTF. Shaded area below the
OTF and above the neural threshold is the area of visibility specified in the
definition of metric AreaOTF.
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 frequency 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.
The area of visibility in this context is the region
lying below the radially averaged MTF and above the neural contrast threshold
function (Charman & Olin, 1965; Mouroulis, 1999). The normalized metric is computed
as
where TN is the neural contrast
threshold function, which equals the inverse of the neural contrast sensitivity
function (Campbell & Green, 1965). When
computing area under rMTF, phase-reversed segments of the curve count as
positive area (Figure A-8). This is consistent
with our definition of SFcMTF as the highest frequency for which rMTF exceeds
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 volume under the MTF have been used in studies of chromatic
aberration (Marcos, Burns, Moreno-Barriusop, & Navarro, 1999) and visual instrumentation (Mouroulis, 1999).
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
where TN is the neural contrast
threshold function defined above. Since the domain of integration extends only
to the cutoff spatial frequency, phase-reversed segments of the curve do not
contribute to area under rOTF. This is consistent with our definition of SFcOTF
as the lowest frequency for which rOTF is below neural theshold. This metric
would be appropriate for tasks in which phase reversed modulations (spurious
resolution) actively interfere with
performance.
The Strehl ratio is often computed in the frequency
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 volume, in the
2-dimensional case) under its Fourier transform. Since the OTF is the Fourier
transform of the PSF, we may conclude that the volume under the OTF is equal 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 this calculation. Although popular, this method lacks rigorous
justification because MTF=|OTF|. This non-linear transformation destroys the
Fourier-transform relationship between the spatial and frequency domains that is
the basis of the central ordinate theorem, which in turn is the justification
for computing Strehl ratio in the frequency
domain.
Strehl ratio computed by the MTF method is
equivalent to the Strehl ratio for a hypothetical PSF that is well-centered with
even symmetry computed as the inverse Fourier transform 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).
The Strehl ratio computed by the OTF method will
accurately compute the ratio of heights of the PSF and a diffraction-limited PSF
at the coordinate origin. However, the peak of the PSF does not necessarily
occur at the coordinate origin established by the pupil function. Consequently,
the value of SROTF is not expected to equal SRX exactly, except in those special
cases where the peak of the PSF occurs at the coordinate
origin.
This metric is similar to the MTF method of computing
the Strehl ratio, except that the MTF is weighted by the neural contrast
sensitivity function
CSFN,
In so doing, modulation in spatial frequencies
above the visual cut-off of about 60 c/deg is ignored, and modulation near the
peak of the CSF (e.g. 6 c/deg) is weighted maximally. It is important to note
that this metric gives weight to visible, high spatial-frequencies employed in
typical visual acuity testing (e.g. 40 c/deg in 20/15 letters). Visual 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 Fourier transform of MTF (which implicitly assumes. PTF=0). Thus, in
general, VSMTF is only an approximation of the visual Strehl ratio computed in
the spatial domain (VSX).
This metric is similar to the OTF method of computing
the Strehl ratio, except that the OTF is weighted by the neural contrast
sensitivity function CSFN,
This metric differs from VSX by emphasizing
image quality at the coordinate origin, rather than at the peak of the PSF.
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.
Since the MTF ≥ the real part of the OTF,
this ratio is always ≤ 1. Creation of this metric was inspired by a
measure of orientation bias of the receptive fields of retinal ganglion cells
introduced by Thibos & Levick (Exp. Brain Research, 58:1-10, 1985).
This metric is intended to quantify the
visually-significant phase shifts in the image. It does so by weighting the MTF
and OTF by the neural contrast sensitivity function before comparing the volume
under the OTF to the volume under the MTF.
The wavefront aberration function is a monochromatic
concept. If a source emits polychromatic light, then wavefront aberration maps
for each wavelength are treated separately because lights of different
wavelengths are mutually incoherent and do not interfere. For this reason, the
definition of metrics of wavefront quality do not generalize easily to handle
polychromatic light. Nevertheless, it is possible to compute the value of a
given metric for each wavelength in a polychromatic source and then form a
weighted average of the results,
where the weighting function V(l) is the
luminous efficiency function that describes how visual sensitivity to
monochromatic light varies with wavelength l.
To the contrary, polychromatic metrics of image quality
for point objects are easily defined by substituting polychromatic images for
monochromatic images. For example, the polychromatic luminance point-spread
function PSFpoly is a weighted sum of the monochromatic spread functions PSF(x,y,l),
Given this definition, PSFpoly may
be substituted for PSF in any of the equations given above to produce new,
polychromatic metrics of image quality. In addition to these luminance metrics
of image quality, other metrics can be 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 compared to the chromaticity
coordinates of each point in the retinal PSF and metrics devised to summarize
the differences between image chromaticity and object chromaticity. Such metrics
may prove useful in the study of color
vision. Given the polychromatic point-spread
function defined above in Equation A-36, a polychromatic
optical transfer function OTFpoly may be computed as the Fourier
transform of PSFpoly. Substituting this new function for OTF (and its
magnitude for MTF) in any of the equations given above will produce new metrics
of polychromatic image quality defined in the frequency domain. Results obtained
by these polychromatic metrics will be described in a future
report.
Since all of the metrics of
wavefront and image quality defined above are intended to measure the optical
quality of an eye, they are expected to be statistically correlated. To 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 Aberration Study (Thibos, Hong, Bradley, & Cheng, 2002). The correlation matrix for all 31 metrics of
optical quality is shown in Figure A-10. Only a
few correlations were found to be statistically insignificant (a=0.05)
and these were coded as zero in the figure.
Figure A-10.
Correlation matrix. Each square indicates the correlation coefficient for the
corresponding pair of metrics. Color bar indicates scale of correlation
coefficient. Ordering of metrics is given in Table 1.
Given the strong correlations between metrics evident
in Figure
A-10, the question arose whether it would be possible to discover a smaller
set of uncorrelated variables that could adequately account for the individual
variability in our study population. To answer this question we used principal
component (PC) analysis (Jackson, 1991). This
analysis indicated that a single PC with the largest characteristic root can
account for 65% of the variance between eyes. This first PC is an orthogonal
regression line that is a “line of best fit” since it provides the
best account of between-eye variation of the individual metrics. For our normal,
well-corrected eyes the first PC gave approximately equal weighting to all 31
metrics except VOTF (which had about 1/4 the weight of the other metrics). This
implies that inter-subject variability of metric values (relative to the mean)
is about the same for each metric yet each metric emphasizes a different aspect
of optical quality. The sign of the weighting was positive for metrics that
increase as optical quality increases and negative for metrics that decrease as
optical quality increases. Thus, PC#1 may be interpreted as an overall metric of
optical quality.
One practical use of principal component analysis in
this context is to identify unusual eyes for which the various metrics of
optical quality do not agree. This outlier analysis is done formally by
computing Hotelling’s T-squared statistic for each eye and flagging those
values that are abnormally large (Jackson, 1991).
This research was supported by National Institutes of
Health Grant EY-05109 (LNT) and EY 08520 (RAA). We thank the authors of the
Indiana Aberration Study for access to their aberration database. We thank.
Jacob Rubinstein for help with differential geometry issues, Xu Cheng for
critical comments and help with figure production, and Austin Roorda for
suggesting the through-focus virtual-refraction paradigm.
Commercial relationships: Thibos and Applegate have a
proprietary interest in the development of optical metrics predictive of visual
performance.
Corresponding author: Larry Thibos.
Email: thibos@indiana.edu.
Address: Indiana University, School of Optometry,
Bloomington, IN, USA
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