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LABORATORY SCIENCE
Goodness-of-prediction of Zernike polynomial
fitting to corneal surfaces
Michael K. Smolek, PhD, Stephen D. Klyce, PhD
PURPOSE: To determine the goodness-of-prediction of the fitting routine by measuring the difference
between topographic corneal surfaces and their Zernike reconstructions as a function of polynomial
order and optical zone size for various corneal conditions.
SETTING: Corneal research laboratory in a university eye center.
METHODS: Corneal topography maps (N Z253) were obtained from the Louisiana State University
Eye Center. A variety of corneal conditions were used: normals; astigmatism; laser in situ keratomileu-
sis, photorefractive keratoplasty (PRK), and radial keratotomy (RK) postoperative cases (myopic spher-
ical corrections); keratoconus suspect; mild, moderate, and severe keratoconus; pellucid marginal
degeneration; contact lens–induced corneal warpage; and penetrating keratoplasty. The root-
mean-square (RMS) error of the goodness-of-prediction of the Zernike representation of corneal sur-
face elevation was extracted for 4, 6, and 10 mm optical zones, whereas Zernike radial orders were
varied from 3 to 14 in 1-order steps. The mean GSEM of the RMS error was plotted as a function
of Zernike order and compared with criteria for normal surface fits.
RESULTS: Fitting accuracy improved as more Zernike terms were included, but some conditions
showed significant errors (when compared with normal surfaces), even with many added terms. For
a 6 mm optical zone, the normal cornea group had the lowest RMS error and did not require terms
above the 4th order to achieve <0.25 mm RMS error. Astigmatism met the 0.25 mm threshold at the
5th order, whereas keratoconus suspect required 7 orders. Laser in situ keratomileusis and PRK met
the 0.25 mm threshold at the 8th order, whereas RK required 10 orders. Contact lens–induced corneal
warpage and mild keratoconus needed 12 orders to meet the 0.25 mm threshold, whereas pellucid
marginal degeneration, moderate and severe keratoconus, and keratoplasty categories were not
well fitted even at 14 orders.
CONCLUSIONS: A 4th-order Zernike polynomial appeared reliable for modeling the normal cornea,
but using a 4th-order fitting routine with an abnormal corneal surface caused a loss of fine-detail shape
information. As more Zernike terms were added, the accuracy of the fit improved, and the result ap-
proached the minimum error found with normal corneas. Unless sufficient higher-order Zernike terms
are included when analyzing irregular surfaces, some diagnostic applications of Zernike coefficients
may not be rigorous. This conclusion also suggests that wavefront shape analysis is similarly depen-
dent on the number of orders used. Current surgical corrections based on normal-eye wavefronts
may fail to capture all visually relevant aberrations in abnormal eyes, such as those having laser retreat-
ments or experiencing corneal warpage from contact lens wear. A clinical goodness-of-fit or goodness-
of-prediction index would indicate whether the number of terms in use has fully accounted for all of
the visually significant aberrations present in the eye.
J Cataract Refract Surg 2005; 31:2350–2355 Q2005 ASCRS and ESCRS
Zernike analysis is used commonly in ophthalmology to ex-
press ocular wavefront error in the form of a polynomial
equation.
1
Specifically, the Zernike polynomial describes
the shape contribution made by a series of sine and cosine
basis functions to the total wavefront error surface. The
phase content of these basis functions varies systematically
with respect to meridian angle and radial direction on
a unit circle (typically defined by the pupil aperture). The
Q2005 ASCRS and ESCRS
Published by Elsevier Inc.
0886-3350/05/$-see front matter
doi:10.1016/j.jcrs.2005.05.025
2350
J CATARACT REFRACT SURG - VOL 31, DECEMBER 2005
individual polynomial terms, as well as certain combina-
tions of these terms, in the Zernike polynomial equation
signify the magnitude and orientation of various aberra-
tions such as defocus, astigmatism, coma, trefoil, and
spherical aberration, along with higher-order aberrations
(HOAs) that have no specific names. As a fitting routine,
the Zernike polynomial is used to analyze not only the
shape of wavefront error but also other complex surfaces,
including the anterior corneal surface.
2,3
This piecewise
method of defining a complex surface shape allows for ad-
ditional higher-order terms to be added to the polynomial
expression as needed. Unfortunately, it is difficult to
know a priori when more higher-order terms are required
to fully model a surface shape. The higher-order terms
should play a significant role in the optical quality of a sur-
face if the collective magnitude of these additional coeffi-
cients is large.
The purpose of this report is to evaluate the ability of
the Zernike polynomial to reproduce corneal shapes of
varying complexity. The results of this study are of interest
because it has been suggested that Zernike analysis may be
applicable toward the development of corneal topography
diagnostic testing tools (eg, Zernike coefficients as inputs
into corneal classification neural networks). A polynomial
approach to diagnostics may replace or supplement the cur-
rently used corneal indices included with many topography
devices. In addition, the results of this study may be ex-
tended more broadly toward the potential limitations of
Zernike fitting when applied to any complex surface shape,
such as the ocular wavefronts of aberrated eyes. Many
surgeons wish to use customized wavefront-guided laser
treatments on abnormal conditions with complex corneal
shapes, such as keratoconus, keratoplasties, warped cor-
neas, and refractive surgery retreatment cases. They argue
that patients with these conditions might benefit most
from corneal reshaping. Irrespective of the concerns about
the appropriateness of refractive surgery on nonnormal
eyes, there is a growing concern among surgeons that the
Zernike fitting method itself may be inaccurate in abnormal
conditions.
It is not known how many terms are sufficient to
achieve an acceptable error level in the Zernike reconstruc-
tion for any given corneal condition. Little or nothing may
be known about the extent and nature of the shape aberra-
tions that are present in any given cornea before an exam-
ination. It is therefore impossible to know ahead of time
whether sufficient numbers of polynomial terms (or or-
ders) will be extracted for a complete Zernike analysis
and accurate reconstruction of the modeled surface. Are 5
polynomial orders enough, or are 8 or possibly 10 orders
needed to record all relevant surface features for a given
corneal condition? This information is particularly needed
when selecting the numerical inputs for neural network–
based diagnostic software such as corneal classification
and condition severity grading utilities.
To address this concern, the goodness-of-prediction of
the fit between Zernike-reconstructed and topographically
derived corneal surface elevation was measured as a func-
tion of Zernike order and optical zone size for a variety of
corneal conditions with different types and amounts of ab-
erration. Goodness-of-prediction is a statistical definition
that compares the actual values of variables with their
mathematically predicted values (such as in Zernike fit-
ting), usually in terms of root-mean-square (RMS) error.
In our study, a higher RMS error in the goodness-of-predic-
tion indicates a poorer representation of the topographically
derived corneal surface elevation by the Zernike-based
reconstruction model.
MATERIALS AND METHODS
TMS-1 (Tomey, Inc) corneal topography maps were obtained
from patient history records at the Louisiana State University Eye
Center. Research use of corneal topography has Institutional Re-
view Board approval at the Louisiana State University Eye Center,
and the research adhered to thetenets of the Declaration of Helsinki.
Corneal conditions that were analyzed included the following
categories: normal sphere (!0.5 diopter [D] of cylinder error);
astigmatism (O0.5 D of cylinder error); laser in situ keratomileu-
sis (LASIK), photorefractive keratectomy (PRK), and radial kera-
totomy (RK) postoperative maps (all refractive surgery maps
were myopic spherical corrections); keratoconus suspect
(KCS); mild, moderate, and severe keratoconus (KC); pellucid
marginal degeneration; contact lens–induced corneal warpage;
and penetrating keratoplasty (PKP). The number of cases in
each category is shown in Table 1. Only maps with noninterpo-
lated mire images were used.
Version 4.07 of CTView software (Sarver and Associates) was
used to analyze the topography maps for elevation fit errors using
corneal optical zones set at 4, 6, and 10 mm diameter zones, with
the radial order of the polynomial terms ranging from 3 to 14 in
1-order steps (ie, 10 to 120 terms). All other CTView settings
were used at their default values. After processing all examina-
tions at the different orders and optical zones, the mean GSEM
of the RMS error for the elevation fit was computed for each con-
dition and plotted as a function of Zernike order number.
Accepted for publication May 18, 2005.
From the Louisiana State University Eye Center, New Orleans,
Louisiana, USA.
Supported by the National Institutes of Health, Bethesda, Mary-
land, USA, and NIH grants R01-EY014162 (Dr. Smolek), R01-
EY003311 (S.D.K.), and P30-EY002377 (L.S.U.).
Neither author has a financial or proprietary interest in any mate-
rial or method mentioned.
Reprint requests to Michael K. Smolek, PhD, Louisiana State Uni-
versity Eye Center, 2020 Gravier Street, Suite B, New Orleans, Lou-
isiana 70112, USA. E-mail: msmole@lsuhsc.edu.
LABORATORY SCIENCE: ZERNIKE FITTING TO CORNEAL SURFACES
J CATARACT REFRACT SURG - VOL 31, DECEMBER 2005 2351
It is important to understand that the RMS error for the ele-
vation fit is not the same measurement as the RMS error of the true
corneal surface or the topographically generated corneal surface,
but is the difference between the topographically computed sur-
face and the Zernike model created from that topographic data.
The study was designed to evaluate the accuracy of the Zernike
polynomial in fitting the corneal surface shape data after record-
ing by the corneal topographer. The study did not measure the ac-
curacy with respect to the true corneal surface, which can never be
exactly known for different clinical conditions. Therefore, this
protocol was designed to isolate and test only the Zernike fitting
algorithm without regard for systematic and random instrument
measurement errors, which might bias the assessment of the Zer-
nike algorithm. Instrument measurement noise is a separate issue
from algorithm fitting errors, and it was reported to be negligible
in a recent study.
4
The protocol in this study also allowed testing
of the fitting algorithm with realistic data from a variety of differ-
ent corneal conditions that are seen in clinical situations.
To assess the accuracy of the polynomial fit, a surface-fit RMS
error of 0.25 mm was chosen as a standard criterion value to com-
pare the magnitude of error across all corneal categories and was
indicated in the graphs by a horizontal red line. The value of
0.25 mm is slightly better than the fitting error measured for the
normal category at a 3rd-order expansion for a 4 mm optical
zone. Because we are examining a mathematical fitting error, the
value of a good fit will be very close to 0, thus necessitating an ex-
tremely precise criterion. An even more strict surface fit RMS error
criterion value of 0.10 mm is displayed on the graphs as a green
line, and this error level approximates that achieved by the normal
category at a 5th-order expansion for a 4 mm optical zone. Achiev-
ing the criterion means that the majority of cases tested in a given
category were below the threshold (ie, the upper limit of the error
bar was at or below the criterion value).
RESULTS
The greater the irregularity of the cornea or the lower
the order of Zernike terms used for fitting, the higher the
RMS error measured between the Zernike fitted model
and the original corneal surface. This relationship was non-
linear. Figure 1,Aand B, shows the surface fit error for the
4 mm optical zone. The normal cornea results are plotted in
both graphs for comparison. Figure 1,Cand D, indicates
the results for a 6 mm optical zone, whereas Figure 1,E
and F, indicates the results with a 10 mm optical zone.
The normal cornea category tended to achieve the lowest
surface-fit RMS error for each of the 3 optical zones at all
fitting orders. The only exception to this rule was a slightly
lower error by the astigmatism group (CYL) at the 3rd
order with a 10 mm zone. (The 3rd-order data points are
plotted off the top of the vertical scale of Figure 1,Eand F).
All surgical conditions except PKP achieved the
0.25 mm criterion threshold by the 4th order when using
a 4 mm zone; however, most surgical conditions needed
10 or 11 orders to achieve the 0.10 mm criterion, which is
a notable difference from the normal group (Figure 1,A).
Up to 8 to 10 Zernike orders were needed to achieve the
0.25 mm error threshold for many of the diseased conditions,
whereas only KCS achieved the 0.10 mm criterion by the
11th order (Figure 1,B). Severe keratoconus (KC3) failed
to reach the 0.25 mm criterion, even at 14 Zernike orders.
Using a 6 mm optical zone, the normal group achieved
the 0.25 mm threshold at the 4th order instead of the third
order, as seen with the 4 mm zone. The PRK and LASIK
groups required 8 Zernike expansion orders to achieve
the 0.25 mm threshold; however, the RK group required
10 orders. None of the conditions was able to achieve the
0.10 mm criterion by the 14th order, with the exception
of the normal group (Figure 1,C). The highly aberrated
PKP condition never achieved the 0.25 mm threshold crite-
rion, even at 14 Zernike expansion orders. In Figure 1,D,
the KCS group achieved the 0.25 mm threshold at approx-
imately 7 orders, whereas KC1 and contact lens–induced
corneal warpage needed 12 orders. Pellucid marginal de-
generation, KC2, and KC3 did not meet the 0.25 mm crite-
rion, even at 14 orders. Only the normal group achieved the
0.10 mm criterion, at approximately 13 orders.
With a 10 mm optical zone, none of the refractive sur-
gery cases on average achieved the 0.25 mm error criterion,
even at 14 Zernike orders, whereas the normal and astigma-
tism error groups needed 9 orders. None of the categories,
including the normal group, could meet the 0.10 mm crite-
rion at the 14th order (Figure 1,E). The KCS group on av-
erage achieved the 0.25 mm threshold at approximately 11
Zernike expansion orders (Figure 1,F). Surprisingly, the
contact lens–induced corneal warpage group showed a fit-
ting error comparable in magnitude to errors in ectasia
conditions (KC1, KC2, KC3, and pellucid marginal degen-
eration), and it never achieved the 0.25 mm threshold even,
at 14 Zernike orders.
DISCUSSION
There are concerns about reliability, precision, and ac-
curacy when using Zernike analysis as a diagnostic tool.
The accuracy of Zernike analysis depends on a number of
Table 1. Study categories.
No. of Cases Corneal Condition
24 Normal (!0.5 D of cylinder)
24 Astigmatism (O0.5 D of cylinder)
59 Keratoconus 22 mild, 22 moderate, 15 severe
19 Keratoconus suspect
68 Myopic refractive surgery
(22 RK, 22 PRK, 24 LASIK)
23 Contact lens–induced corneal warpage
16 Penetrating keratoplasty
20 Pellucid marginal degeneration
253 Total
LASIK Zlaser in situ keratomileusis; PRK Zphotorefractive keratectomy;
RK Zradial keratotomy
LABORATORY SCIENCE: ZERNIKE FITTING TO CORNEAL SURFACES
J CATARACT REFRACT SURG - VOL 31, DECEMBER 20052352
Figure 1. Surface fit RMS error as a function of Zernike expansion order. A: 4 mm optical zone, surgical conditions. B: 4 mm optical zone, disease, and dis-
orders. C: 6 mm optical zone, surgical conditions. D: 6 mm optical zone, disease, and disorders. E: 10 mm optical zone, surgical conditions. F: 10 mm optical
zone, disease, and disorders.
LABORATORY SCIENCE: ZERNIKE FITTING TO CORNEAL SURFACES
J CATARACT REFRACT SURG - VOL 31, DECEMBER 2005 2353
factors, such as the irregularity of the surface and the num-
ber of polynomial terms or orders used to fit the surface.
3–7
Using too few terms with a highly irregular surface causes
the Zernike fit (ie, the Zernike-based model of the surface)
to be overly smooth and inaccurate when compared
with the actual surface. In a recent study, a patient’s best
spectacle-corrected visual acuity (BSCVA) was significantly
correlated to corneal surface elevation when represented by
both 4th-order and 10th-order Zernike polynomials; how-
ever, this same study showed that BSCVA was significantly
and equally well correlated to the difference between the
Zernike model surface and the corneal surface.
3
In other
words, the residual part of the corneal surface shape that
was beyond the 4th-order and 10th-order Zernike analysis
was just as significant to visual acuity as the part included
by the Zernike polynomial. This is not an unexpected result
given the way Zernike basis functions isolate and decom-
pose individual surface features of the complete surface. Ir-
regular astigmatism from HOAs is expected to be caused by
high spatial frequency surface shape features on the cornea,
and likewise, the HOAs degenerate the high spatial fre-
quency content of the formed retinal image.
Unfortunately, HOAs tend to be decomposed and
spread out by Zernike analysis into numerous individual
components that can have virtually no clinical relevance
alone. In particular, higher-order Zernike terms with high
angular frequency (terms along the outer sides of the Zer-
nike pyramid) are highly artificial shapes, so it is not sur-
prising to learn that they have less impact on vision.
8,9
Although it may be impossible to show a statistically signif-
icant correlation between any given higher-order Zernike
term and a visual deficit, the HOAs from individual terms
are collectively significant in their total effect on vision.
So, higher-order Zernike coefficients may still be relevant
in terms of diagnostic testing, as long as enough orders
are used for a given condition.
Although much effort has been directed toward deter-
mining the wavefront aberrations in normal eyes,
10–13
it
has been difficult to estimate how many Zernike orders
are needed to represent accurately the shape aberrations
of the cornea for various conditions. The results in the cur-
rent study provide a better understanding of the relative
clinical significance of corneal aberrations for different
conditions with different optical zones, at least in terms
of the topographic shape of the cornea. This study now pro-
vides a starting point from which we can design Zernike-
based diagnostic tools that may someday supplement or
replace the corneal research indices now in use. It is
now clear, for example, that limiting the Zernike analysis
to only several orders may cause incorrect assessments of
the severity of more advanced stages of keratoconus. One
of the surprising findings in our study was the signifi-
cantly large amount of residual shape aberrations in
many corneal conditions that simply could not be ac-
counted for by using only the low-order Zernike polyno-
mials, even though the bulk of the aberrations are
contained in these low-order terms. Although not directly
comparable, these latest results nevertheless suggest that
surgical laser systems that use only 4th-order or 5th-order
Zernike-based wavefront models might not fully correct
wavefront aberrations in eyes with either previous refrac-
tive surgery or a disorder that affects the optics of the eye.
This implication holds true for any abnormal eye, includ-
ing cases of contact lens–induced corneal warpage and
keratoconus suspects that are becoming more likely to
be candidates for refractive surgery, even though most sur-
geons still consider these conditions contraindications to
refractive procedures. In addition, it may be possible for
even clinically normal corneas to have their aberrations
incompletely recorded if the Zernike polynomial is limit-
ed to low orders and a large optical zone is considered, as
shown in Figure 1,E. By using a large optical zone or pu-
pil, HOAs become increasingly significant even in normal
eyes.
An important technical issue underlying the applica-
tion of higher-order Zernike terms to refractive surgery is
the physical limitation imposed by the diameter and energy
profile of the laser beam and the amount of tissue that can
be ablated in a single pulse.
14
Narrower beams may allow
a higher resolution for achieving a correction, but mini-
mum beam diameter is limited by diffraction effects. Finer
beams also require more precise tracking and more time to
ablate tissue, and these 2 factors tend to compete against
one another in terms of efficiency and accuracy. Yet another
technical issue involved in Zernike modeling accuracy is
the spatial resolution of the wavefront sensor and the ability
to maintain registration between the Zernike ‘‘map’’ and the
eye. Many current wavefront sensors fail to capture all rel-
evant information contained within a wavefront, which
compounds the problem of mathematical smoothing in-
duced by the fitting algorithm. All these issues are beyond
the scope of the current study, but they are worthy of fur-
ther research because they are a growing concern to clini-
cians regarding the efficiency, reliability, and accuracy of
refractive surgery in the correction of HOAs. Of course,
even beyond the technical issues of ablation and sensors,
biological and biomechanical tissue responses also play
a role in the accuracy and precision of wavefront-guided
laser surgery.
Whereas adding more Zernike terms will tend to im-
prove the fitting accuracy for abnormal corneas, current la-
ser systems and aberrometers do not necessarily allow more
Zernike terms to be included routinely into the calculation
for a more accurate correction. More fundamentally, it is
uncertain how many more terms need to be added with
any given eye to achieve an accurate fit. Our concern about
LABORATORY SCIENCE: ZERNIKE FITTING TO CORNEAL SURFACES
J CATARACT REFRACT SURG - VOL 31, DECEMBER 20052354
the accuracy of the Zernike fitting routine for complex
shapes has been echoed by ophthalmologists working inde-
pendently on the limitations of Zernike fitting as it applies
to ocular wavefronts. For example, potential limitations in
Zernike fitting for corneal ablation algorithms have been
discussed by refractive surgeons Douglas Koch (‘‘Beyond
Zernikes: Improving the Basis for Wavefront Ablations
with New Algorithms,’’ presented at the ISRS/AAO, 2003)
and Julian Stevens (‘‘Therapeutic Applications of Custom
Corneal Ablation,’’ presented at the ISRS/AAO 2003 and
‘‘Fourier Transform Wavefront-Guided Therapeutic Cus-
tom Corneal Ablation,’’ presented at the ASCRS Sympo-
sium on Cataract, IOL and Refractive Surgery, San Diego,
California, USA, May 2004). The Fourier approach can ac-
curately fit corneal and wavefront error surfaces in even
highly aberrated eyes (with the proper high spatial resolu-
tion sensor), and the result can be used to guide lasers dur-
ing customized corneal ablation procedures. It remains to
be seen whether the Fourier fitting method outperforms
the Zernike fitting method in all clinical situations, but it
does appear to be more efficient. One additional advantage
is that the Fourier transform also allows direct computation
of the point-spread function, rather than calculating an
overly smooth point-spread function from the already
smooth Zernike polynomials used to fit the original wave-
front.
15,16
A comparison of the similarities and differences
between Zernike and Fourier transform methods is well
beyond the scope of this report, but it is a topic worthy of
further investigation.
We have shown that the accuracy of Zernike fitting of
corneal topography surface elevation data depends on the
number of radial orders used in the fit; the more terms ap-
plied, the closer the RMS error approaches 0 (a perfect fit).
The accuracy of the fit is highly dependent on the corneal
condition, and this becomes an important concern when
applying Zernike analysis to corneal diagnostics. Corneal
diagnostic applications that are now under development,
and that use Zernike-based shape analysis, may be inac-
curate unless sufficient numbers of Zernike orders are
extracted during the analysis. Even relatively smooth post-
refractive surgery corneas may have substantial higher-
order irregularities that require many Zernike terms to
represent accurately the surface shape. Because of the close
interrelationship between corneal shape and total ocular
wavefront error, we believe that the results of this study
also suggest potential limitations in surgically correcting
refractive errors with present-day wavefront-guided laser
systems using Zernike fitting, except for corneas that
have few HOAs.
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