Applied Mathematics, 2013, 4, 1694-1701
Published Online December 2013 (http://www.scirp.org/journal/am)
http://dx.doi.org/10.4236/am.2013.412230
Open Access AM
Optimization of Linear Filtering Model to Predict
Post-LASIK Corneal Smoothing Based on Training Data
Sets
Anatoly Fabrikant, Guang-Ming Dai, Dimitri Chernyak
Research and Development, Abbott Medical Optics, Milpitas, USA
Received August 29, 2013; revised September 29, 2013; accepted October 6, 2013
Copyright © 2013 Anatoly Fabrikant et al. This is an open access article distributed under the Creative Commons Attribution Li-
cense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
Laser vision correction is a rapidly growing field for correcting nearsightedness, farsightedness as well as astigmatism
with dominating laser-assisted in situ keratomileusis (LASIK) procedures. While the technique works well for correct-
ing spherocylindrical aberrations, it does not fully correct high order aberrations (HOAs), in particular spherical aberra-
tion (SA), due to unexpected induction of HOAs post-surgery. Corneal epithelial remodeling was proposed as one
source to account for such HOA induction process. This work proposes a dual-scale linear filtering kernel to model such
a process. Several retrospective clinical data sets were used as training data sets to construct the model, with a downhill
simplex algorithm to optimize the two free parameters of the kernel. The performance of the optimized kernel was
testedon new clinical data sets that were not previously used for the optimization.
Keywords: Simulation-Driven Optimization; Downhill Simplex Method; Corneal Smoothing; LASIK
1. Introduction
Historically, eyeglasses and contact lenses have been
used to alleviate refractive problems such as nearsight-
edness, farsightedness, and astigmatism. With the advent
of excimer lasers [1] specially designed for laser-assisted
in situ keratomeliusis (LASIK) and photorefractive ker-
atectomy (PRK), patients started to enjoy a new type of
vision correction that is free of eyeglasses. With wave-
front-guided LASIK [2], the correction of ocular aberra-
tions is no longer limited to the so-called low-order ab-
errations, i.e., the spherocylindrical error that can be cor-
rected with traditional eyeglasses. This new technology
enables the correction of higher-order aberrations (HOAs)
that are beyond the spherocylindrical error, most notably
spherical aberration and coma. Thus, super sharp vision
is attainable in theory with the wavefront-guided LASIK.
Unfortunately, the human cornea is not a piece of plas-
tic [3]. With LASIK, it involves first cutting a flap on the
corneal stroma, lifting it to the side, then delivering the
UV laser pulses to remove tissue, and finally putting
back the flap, which heals shortly after surgery. The pre-
cise design of an ablation target may cut the corneal
stroma as needed to achieve a desired shape immediately
after surgery. However, the biomechanical process and
the corneal epithelial remodeling after surgery change the
surface of the cornea, resulting in deviations from the
original optical design of the ablation shape. Therefore,
the post-operative induction of HOAs, especially sphere-
cal aberration (SA), is currently among the most serious
challenges for laser vision correction technology. Among
several possible root causes of SA induction, the post-
operative cornea remodeling was found the most impor-
tant [4]. The main effect of the cornea remodeling is the
smoothing of epithelial anterior surface, when the epithe-
lium tends to grow thicker at the center and fill in the
dips of the cornea surface, created by refractive surgery
[5]. The epithelial smoothing causes some spherocylin-
drical regression after refractive surgery, which can be
corrected by a linear adjustment of the intended refrac-
tive correction. It also leads to the induction of high-or-
der aberrations, which are increasingly strong for high
myopia and hyperopia cases [4,6].
Among the HOAs induced, spherical aberration is the
most significant. In general, the amount of the SA induc-
tion tends to increase with post-surgery time. Several
months after surgery when the cornea stabilizes, the in-

A. FABRIKANT ET AL. 1695
duced SA shows a statistically significant trend versus
the magnitude of the treated refraction. Figure 1 shows
the post-operative SA over a 6 mm diameter as a func-
tion of the pre-operative manifest refraction in spherical
equivalent (MRSE). The regression slope of the induc-
tion is remarkably consistent between different data sets.
The purpose of this study is to find a corneal smooth-
ing model to represent the corneal change post-surgery
using an optimization algorithm, based on retrospectively
available clinical data. The kernel is then tested with
other clinical data sets that were not previously used for
the optimization. This well tested kernel can then be used
to “reverse” the biological corneal smoothing effect by a
mathematical deconvolution process. An improved treat-
ment algorithm can then be designed, which hopefully
will remove the induced spherical aberration.
2. Modeling of Post-Operative Corneal
Smoothing
Various models can capture geometric changes to the
surface of the human cornea occurring after the surgery.
We considered an optimized linear filter (OLF) model,
which describes post-operative smoothing of the corneal
ablation. This model is characterized by a small set of
parameters determined by a model optimization based on
retrospective clinical data.
The post-operative epithelial smoothing process can be
simulated by means of a simple mathematical model.
This model defines the shape of the post-operative cor-
nea surface as a convolution of the ablation target profile
with a linear smoothing filter as
 
post-op pre-op,hhKxyTx 
where h stands for the elevation maps of the corneal sur-
face for pre-operative and post-operative situations, re-
spectively,

denotes a convolution operation, T(x,y) is
the ablation target profile and K(x,y) is the linear smooth-
ing filter kernel. A simple squared Butterworth low-pass
filter [7] has been proposed. A squared Butterworth filter
of the first order takes a form with the square term of the
spatial frequency as

2
2
1
,
1
xy
r
Kk kk
s


(2)
where K(kx,ky) is the Fourier transform of K(x,y),
22
rx
kkk
y

, and s is a parameter representing the
scale of the kernel. With limited success using the
squared Butterworth filter as defined by Equation (2), we
began consideration of dual-scale and triple-scale OLFs
that has a somewhat similar shape as the squared But-
terworth filter. Our tests show that a dual-scale OLF
model has the advantage of faster convergence and pro-
per account of biological change of the epithelial cells
than the triple-scale model. Therefore, we have used a
dual-scale OLF kernel that is defined as

2
24
1
,
1
Kxy
rr
ss
4







(3)
where 22 rxy is the radial distance from the co-
ordinate origin, s2 and s4 are two unknown free parame-
ters to be determined. For our application, r, s2 and s4 all
have dimensions in mm. Figure 2 shows the cross-sec-
tion of the kernel and its power spectrum.

,y
(1)
DS1: y = -0.0306x + 0.0484
R² = 0.3647
DS2: y = -0.0373x - 0.018
R² = 0.3975
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
-14 -12 -10-8-6-4-20
Post-Operative SA (μm)
Pre-Operative MRSE (D)
Data Set 1
Data Set 2
Figure 1. Post-LASIK spherical aberration (SA) as a function of the pre-operative manifest refraction in spherical equivalent
(MRSE) for two data sets.
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A. FABRIKANT ET AL.
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0
0.05
0.1
0.15
0.2
0.25
-1-0.50 0.5 1
K(r)
Radial distance (mm)
1.E-5
1.E-4
1.E-3
1.E-2
1.E-1
1.E+ 0
-5 -3 -1135
K(r)
Radial distance (mm)
1.E-2
1.E-1
1.E+ 0
1.E+ 1
1.E+ 2
1.E+ 3
1.E+ 4
0510 15 20
|F[K(k)]|
k (cycles/mm)
Figure 2. Cross-section of the optimized linear filter. Left panel, linear scale of the center of the kernel; middle panel, loga-
ithmic scale of the entire kernel; right panel, power spectrum of the kernel. r
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A. FABRIKANT ET AL.
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1697
3. Optimization of the Kernel Parameters “obs” stands for observation, i.e., clinical outcome.
Several different optimization algorithms have been
tested. Our choice of the downhill simplex method [8,9]
works well with our model and the data sets. With four
clinical data sets (two parts of Data Set 1, low myopia
and high myopia, and Data Sets 3 and 5) used for opti-
mization, we found that s2 = 0.0334 mm and s4 = 0.464
mm give the minimum σ as defined in Equation (4).
By using Equation (3) in Equation (1) using the pre-op-
erative and post-operative wavefront data as well as the
treatment targets for various previously treated eyes, the
two unknown parameters s2 and s4 can be obtained by
minimizing the difference between the simulated post-
operative wavefront error and the observed post-opera-
tive wavefront error. This minimization is a least-squares
type which minimizes the regression slopes of the post-
operative spherical equivalent (SE) and post-operative
SA as a function of the pre-operative SE for all eyes as


2
2simu obs
obs
2
simu obs
obs
slopeSE slopeSE
slopeSE
slopeSA slopeSA
slopeSA



























(4)
With these two kernel parameters, application of the
treatment parameters for those eyes using Equation (1)
enables us to obtain the simulated clinical outcome,
which includes spherocylindrical error (wavefront sphe-
rical equivalent, or WSE) and SA. The WSE is measured
in diopters (D) and the SA is measured in microns (µm)
over a 6 mm diameter. Figure 3 shows the comparison
between the observed and simulated post-operative out-
come for two data sets that were used for the optimiza-
tion. Both the post-operative WSE and SA as a function
of the pre-operative WRSE are plotted. With no surprise,
the regression slopes of the simulated eyes agree well
with those of the observed eyes.
where slopeSE and slopeSA are the regression slopes of
the post-operative SE versus pre-operative SE and post-
operative SA versus pre-operative SE, respectively. δ
stands for 95% confidence interval of the observed slope.
Subscript “simu” stands for simulation and subscript
Once the parameters s2 and s4 are determined, the OLF
kernel can be determined based on Equation (3). To ob-
tain a new target shape that is capable of removing the
y = 0.1547x - 0.0464
Observed 2: y = 0.1547x - 0.0464
R² = 0.1638
Simulated 2: y = 0.173x + 0.0247
R² = 0.9054
Observed 1: y = 0.0969x + 0.4002
R² = 0.0395
Simulated 1: y = 0.0956x + 0.3911
R² = 0.6697
-3
-2
-1
0
1
2
3
-14 -12 -10-8-6-4-20
Post-Operative WSE (D)
Pre-Operative MRSE (D)
Observed
Simulated
Observed: y = 0.0594x - 0.1129
R² = 0.0687
Simulated: y = 0.065x -0.2855
R² = 0.0793
-3
-2
-1
0
1
2
3
-12-10-8 -6 -4 -20
Post-Operative WSE (D)
Pre-Operative MRSE (D)
Observed
Simulated
Observed 2: y = -0.0216x + 0.0733
R² = 0.0834
Simulated 2: y = -0.0319x + 0.0131
R² = 0.4038
Observed 1: y = -0.0121x + 0.218
R² = 0.0122
Simulated 1: y = -0.0355x - 0.0041
R² = 0.437
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
-14-12-10-8 -6 -4 -20
Post-Operative SA (μm)
Pre-Operative MRSE (D)
Observed
Simulated
Observation: y = -0.0324x + 0.0096
R² = 0.4012
Simulated: y = -0.0219x -0.0063
R² = 0.2612
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
-14 -12-10-8-6-4-20
Post-Operative SA (μm)
Pre-Operative MRSE (D)
Observed
Simulated
Figure 3. Comparison of simulated and observed post-operative aberrations (WSE and SA) for Data Set 1 (left panels, two
ubsets, n = 390) and Data Set 3 (right panels, n = 76). s

A. FABRIKANT ET AL.
1698
post-operative induction of spherical aberration, a de-
convolution process of Equation (1) can be employed as


newINV current
*
current
22
,
,SNR
xy
xy
TKT
Kk k
F
Kk k












T



(5)
where F(·) stands for a Fourier transform, * denotes a
complex conjugate, Tcurrent is the current treatment target
with induction of post-operative SA, Tnew is the new tar-
get that is expected to remove the post-operative SA, and
KINV is the inverse kernel of K(x,y). This is the typical
Wiener filtering technique [9]. The SNR is used to pre-
vent noise amplification and oscillation at the edge. A
constant value of 0.1 was used for practical purpose.
4. Verification of the Model with New Test
Data
The effect of post-LASIK central corneal thickening
caused by epithelial smoothing has been observed
previouslyin the literature [4,5,10-12] and provided at
least partial explanation for regression after refractive
surgery for myopia. The OLF obtained in this study
confirms the central corneal thickening phenomena, which
gives biological support of the kernel.
For the first verification, we used the kernel optimized
with Data Sets 1, 3 and 5 and applied it to Data Sets 2
and 4, as depicted in Figure 4. It is interesting to see that
the regression slopes of the simulated eyes agree well
with those of the observed eyes, even though these data
sets were not used for the optimization. This result is
expected because the model is supposed to simulate the
post-operative corneal smoothing process, which should
not be different for different data sets.
For the second verification, recall that we used SE and
SA as two aberration parameters for optimization in
Equation (4). More convincingly, we used the same ker-
nel to obtain similar regression slopes for the secondary
spherical aberration, which is not a parameter used in the
optimization, as shown in Figure 5. Again, we have
good matches between the observed and the simulated
slopes. This result comes as expected, as the induction of
HOAs from the corneal smoothing is primarily rotation-
ally symmetric. Secondary spherical aberration is the
most important rotationally symmetric after sphere and
primary spherical aberrations.
Some regression plots show a constant offset between
Observed: y = 0.1146x - 0.0436
R² = 0.1323
Simulated: y = 0.0894x - 0.2302
R² = 0.7088
-3
-2
-1
0
1
2
3
-10-8-6-4-2 0
Observed: y = 0.1063x - 0.3415
R² = 0.0661
Simulated: y = 0.1005x - 0.2776
R² = 0.2762
-3
-2
-1
0
1
2
3
-10-8-6-4-2
Post-Operative WSE (D)
Pre-Operative MRSE (D)
Observed
Post-Operative WSE (D)
Pre-Operative MRSE (D)
Simulated
0
Observed
Simulated
Observed: y = -0.037x + 0.0675
R² = 0.2043
Simulated: y = -0.0445x -0.0296
R² = 0.6803
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
-10-8-6-4-2 0
Observed: y = -0.0475x - 0.0376
R² = 0.301
Simulated: y = -0.0574x - 0.0596
R² = 0.6553
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
-10-8-6-4-2
Post-Operative SA (μm)
Pre-Operative MRSE (D)
Post-Operative SA (μm)
Pre-Operative MRSE (D)
0
Observed
Simulated
Observed
Simulated
Figure 4. Comparison of simulated and observed post-operative aberrations (WSE and SA) for Data Set 2 (left panels, n = 74)
and Data Set 4 (right panels, n = 72).
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A. FABRIKANT ET AL. 1699
Observed: y = -0.0068x + 0.0047
R² = 0.0915
Simulated: y = -0.011x - 0.001
R² = 0.3896
Observed: y = -0.0025x + 0.0308
R² = 0.0059
Simulated: y = -0.011x + 0.0066
R² = 0.3722
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
-12-10-8 -6 -4 -20
Post-Operative secondary SA (μm)
Pre-Operative MRSE (D)
Observed
Simulated
Observed: y = -0.017x - 0.0149
R² = 0.5124
Simulated: y = -0.0157x - 0.0209
R² = 0.6401
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
-10-8-6 -4-2
Post-Operative secondary SA (μm)
Pre-Operative MRSE (D)
0
Observed
Simulated
Observed: y = -0.0125x - 0.0011
R² = 0.3718
Simulated: y = -0.0186x -0.0252
R² = 0.6283
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
-10-8-6-4-2 0
Observed: y = -0.0075x + 0.0022
R² = 0.2231
Simulated: y = -0.0098x - 0.0117
R² = 0.5033
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
-10-8-6-4-2
Post-Operative secondary SA (μm)
Pre-Operative MRSE (D)
Post-Operative secondary SA (μm)
Pre-Operative MRSE (D)
Observed
0
Observed
Simulated
Simulated
Figure 5. Post-operative secondary SA as a function of the pre-operative SE for simulated and observed eyes in the Data Set 1
(upper left, n = 390), Data Set 2 (upper right, n = 74), Data Set 3 (lower left, n = 76), and Data Set 4 (lower right, n = 76). All
eyes are myopic.
the simulated and the observed trend lines. These offsets
for post-operative SE or SA trend are about the same for
all pre-operative MRSE values, indicating that they do
not depend on ablation depth. They may be caused by the
creation of the LASIK flap [10-13]. Depending on the
choice of microkeratome and individual surgeon tech-
nique, the flap-induced aberrations may differ from site
to site or surgeon to surgeon [14].
5. Discussion
Search of the smoothing kernel to model the post-
operative induction of spherical aberration is a relatively
new field of study. Based on the observation of post-op-
erative refractive regression, a simple low-pass squared
Butterworth filter was proposed [7]. This kernel is de-
fined by a single free parameter, which characterizes the
scale of smoothing. Unfortunately, this model does not
provide a satisfactory fitting for the regression slopes for
both post-operative low-order refraction and high order
aberrations simultaneously. Optimization for both refrac-
tion and spherical aberration leads to diverged outcomes.
The proposed OLF with two free parameters models a
dual-scale smoothing. Looking at the cross-section of the
kernel in logarithmic scale in Figure 2, the sharp core
corresponds to the short scale diffusion process and the
wide wings correspond to the long scale smoothing proc-
ess. These two separated processes can be linked to the
post-operative corneal change in low-order and high-
order aberrations, respectively. Consequently, the OLF
yields a good match for both low-order aberrations (WSE)
and high order aberrations (SA) observed clinically. Fur-
thermore, this match can be extended to different data
sets and different aberration types (secondary spherical
aberration).
Looking at the inverse kernel KINV, as depicted in Fig-
ure 6, we found that the peak of the power spectrum of
the inverse kernel corresponds to the size of the superfi-
cial cells of the epithelium. This may not be just a coin-
cidence, as the movement of the epithelial cells, espe-
cially the superficial cells, it’s attributed to the mecha-
nism of the post-operative corneal smoothing. As the
smoothing kernels generally smooth high curvature areas,
the effect of the inverse kernel works exactly the oppo-
site, sharpening areas that have high curvature changes.
The link of the peak of the power spectrum of the inverse
kernel to the size of the superficial cells of the epithelium
rovides another layer of support of our model. p
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A. FABRIKANT ET AL.
1700
-20
0
20
40
60
80
100
-0.5 -0.3-0.10.10.30.5
Kinv(r)
Radial distance (mm)
1.E+1
1.E+2
1.E+3
1.E+4
1.E+5
1.E+6
0510 15 20 25 30
|F[Kinv(k)]|
k (cycles/mm)
Superficial cells
Figure 6. Cross-section of the inverse kernel of the OLF. Left panel, the inverse kernel; right panel, the power spectrum.
Similar to the neural network technique where a data
set is used to train the network and a new data set is used
to test the network, we used combination of data sets to
optimize the free parameters of the proposed kernel and
used new data sets to test the effectiveness of the model.
The successful test of the model strengthens is useful-
ness, which can be important when costly clinical trials
are decided.
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