Download Limbus versus pupil center for ocular alignment

Limbus Versus Pupil Center for Ocular Alignment
Measurement With Corneal Reflexes
Jean-Cyriaque Barry and Andreas Baches
Purpose. To investigate the accuracy of ocular misalignment measurement, using corneal
reflections.
Methods. Corneal reflex positions were measured relative to two landmarks, the limbus center
and the entrance pupil center, using high-resolution digital images for cyclopean gaze angles
from 0° to 18.8° (34.04 prism diopters; PD) to the right and to the left in 10 subjects. Distance
h from the center of the corneal curvature to each landmark was determined from linear
regressions and showed significant differences between both conditions: mean Anmbus was 5.243
mm, and mean /jpupi| was 4.884 mm. From these, limbus- and entrance-pupil-center-related
Hirschberg ratios were determined as lT/mm (19.43 PD/mm) and 11.82°/mm (20.92 PD/
mm), respectively; and ocular alignment was calculated for both conditions. Simulated angles
of strabismus were calculated as the binocular difference between ocular alignment of the
right and left eyes (condition 1), and as the monocular difference between the ocular alignment
of right eyes and left eyes separately (condition 2).
Results. Condition 1: Errors in simulated angles of strabismus were approximately twice as
large for entrance pupil center compared with those in limbus-center-based evaluation; in
the primary position, the 95% pupil-related confidence interval of the binocular difference was
±5.217° (9.1 PD), compared with ±3.174° (5.5 PD) for the limbus-related option. Condition 2:
Errors were approximately equal.
Conclusions. The entrance pupil center is a less reliable landmark than is the limbus center
for measuring ocular alignment by using corneal reflections, because of unreliable positions
of the entrance pupil center; different mean Hirschberg ratios should be used in both conditions. Invest Ophthalmol Vis Sci. 1997; 38:2597-2607.
A . number of corneal reflex techniques for measuring
ocular alignment have been suggested and analyzed.1"21
However, the relative merits of these techniques have,
to our knowledge, never been studied experimentally
by direct comparison with the same alignment data.
Most techniques may, in principle, be derived from
equation 1 5 1 9 by adequate substitutions or approximations:
(f> = arcsin (x/h)
(1)
where
(f>: rotation angle of the corneal axis relative to the
light source,
x: displacement of corneal reflex relative to an axially
located ocular landmark
h: distance of the axially located landmark to the center of the corne; • curvature,
as illustrated in Figure 1. Equation 1 results from a
paraxial approximation, given a cyclopean eye with a
spherical cornea. It holds best for eye rotation angles
that are not large (less than 20°) and in conditions in
which the light source is not close to the cornea—
that is, for distances larger than one third of a meter.19
A simplified relationship uses the Hirschberg ratio
(HR) in a linear approximation of equation 1. That
From the Department of Ophthalmology, RWTH Aachen University, Germany.
Supported in part by DFG (Ef 6/3-2 and Ba 1316/5-3). Corporate support received
from Chiron Adatomed Intraocular Lenses, National Instruments, and XOn
Software, Munich; Management fntelligenter Technologien, Aachen—
Kornelimilnster; Boston Polymer Technologies, Weinheim; and eltec, Mainz,
is:
Germany; Graftek, Mirmande, France; and pro cornea bv, Erbeek, The Netherlands.
Submitted for publication November 25, 1996; revised February 12, 1997; accepted
June 16, 1997.
Proprietary interest category: N.
where
Reprint requests: J-C Barry, Department of Ophthalmology, Medical Faculty of
HR.
Eberhardt Karls University, Schleichstr. 12-16, D-72076 Tubingen, Germany.
77 = HR*x
(2)
77: is the eye rotation angle calculated with the
Investigative Ophthalmology & Visual Science, November 1997, Vol. 38, No. 12
Copyright © Association for Research in Vision and Ophthalmology
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Investigative Ophthalmology & Visual Science, November 1997, Vol. 38, No. 12
varies with ocular components and with the landmark
chosen.
Influence of various parameters on the HR was
reference
examined previously.1213151819 A dependence of HR
on the corneal radius may be predicted from Figure
1, in that the distance h depends on the position of
the center of curvature of the anterior corneal surface,
and hence on its corneal radius r. Because of peripheral flattening of the cornea, the values for the corneal
radius rand the distance h increase; consequently, the
HR decreases in the periphery.21
light source
There is widespread agreement that, within the
limits of measurement accuracy, a single HR of 21 to
22 PD/mm may suffice3'4'6'12"16'18'19 to measure ocular
alignment for clinical and scientific purposes, regardless of individual factors. However, large variations of
FIGURE l. Sketch of how equation 1 (see Methods) can be the individual HR of the order of up to 100% relative
derived in the paraxial approximation. A small light source
to the mean HR have to be expected.5121318
positioned sufficiently far away is imaged by the cornea (apIn securing higher accuracy, little attention has
proximated as a spherical surface) as a virtual image, the
been paid to the role of h or, in other words, to the
anterior (Purkinje I) corneal reflex. It is situated on the axis
proper choice of the reference plane and landmark.
connecting the light source and the center of the curvature
of the cornea, hitting the cornea normally. The corneal reflex Most often, the corneal reflex position was measured
relative to the entrance pupil center. The entrance
displacement x is the projection of the separation between
pupil is the virtual, enlarged image of the physical
the central landmark and the corneal reflex onto the plane
pupil, which appears a little closer to the observer
normal to the observer's imaging axis, which is assumed to
than the physical pupil.23
coincide with the light source direction. The distance x depends on the eye rotation angle 4> and on the axial position
For a quick, clinical estimation of corneal reflex
of the chosen landmark relative to the center of curvature;
position (for example, with the Hirschberg test), the
that is, on the distance, h, as stated in equation 1.
entrance pupil center is probably the only viable landmark relative to which the corneal reflex asymmetry
between both eyes may be judged, in that there is
Equation 2 results from a series expansion22 of
usually a bright, easily observable contrast between the
arcsin(x) and truncation after the linear term, and
entrance pupil and the corneal reflex. The entrance
from the approximation that the cyclopean gaze angle
pupil center is also used as a landmark for ocular
P equals the true eye rotation angle of a real eye that
alignment measurement in recently developed digital
is decentered by half the interpupillary distance. Both
imaging
devices for refractive screening.24"26
approximations have adverse effects: The approximation arcsin(x) « x means that the calculated eye rotaThe limbus margins27 or the center of the limbus
tion angle will be underestimated. The cyclopean eye
were used as alternative landmarks, for instance, beapproximation results in an overestimation of the calcause the margins of the entrance pupil were more
culated eye rotation angle, and because both approxidifficult to locate in the presence of a dark iris than
mations are of the same order (for instance, ~2% for
was the limbus with visible light.3'6 This was combined
P = ±20° [36.4 PD]), they compensate each other,
with computation of eye alignment from ratios of disexplaining to some extent the good results obtained
tances—for instance, the ratio of corneal reflex diswith this approximation.
placement to the width of the limbus.17'20 No calibraThe HR can be gained empirically (HR^pj,.) from
tion for measurement of distances in images would
linear fits of gaze angle P versus measured corneal
then be necessary; consequently, the examination disdisplacement x It may also be calculated (HRcaic) from
tance did not have to be controlled.
equation 1 by setting x = 1 mm for any appropriate
The movement of the limbus or of its center with
chosen or given distance h and dividing the resulting
eye rotation is readily described through the movevalue </>HR by 1 mm.5
ment of the limbus in the normal plane of the observThe determining parameter for the theoretically
er's direction of gaze as h • sine of the eye rotation
derived HRcaic is the distance h that is the separation
angle. The entrance pupil center, however, may unbetween the center of anterior corneal curvature and
dergo a prismatic effect with eye rotation, because of
the reference plane in which the landmark lies, reits refraction through the cornea. Hence, its motion
garding which the decentration x of the corneal reflex
in the normal, plane of the observer's imaging axis
may be specified (Fig. 1). This implies that the HR
would be more difficult to predict.
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Accuracy of Ocular Alignment Measurement
cnnwms
A
pmr of *ye»
wilhPurlunje
1 reflex
f
—
FIGURE 2. Design of apparatus used to record corneal reflex
positions in different gaze directions up to 18.8° of cyclopean gaze angle /? to the right (positivesign) and left (negative
sign). The distance of the central fixation target to the subject's pupillary plane was adjusted to 50 cm.
Therefore, the use of the limbus center as a landmark might offer advantages over the use of the entrance pupil center—more so, because the latter is
known to vary significantly between subjects28 and is
prone to unpredictable asymmetric shifts with pupil
constriction,29 occurring with changes of accommodation, vergence, ambient light levels, or neurovegetative ton us.
Therefore, the object of this study was to examine
to what extent the choice of the limbus or entrance
pupil center as a landmark influences the experimentally found HR, and which of the two choices leads to a
more accurate measurement of the ocular alignment.
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fixation target was used to generate a bright corneal
reflex in both eyes. Two additional units, positioned
8.8 cm to the left and right of the central light source,
were used to generate additional Purkinje images for
further studies.
For each gaze direction, three pictures were taken
and stored. The horizontal positions of Lhe corneal
reflex were measured in 10 young, orthotropic subjects for all gaze directions from 0° (0 PD) to 18.8° to
the right and to the left (Fig. 3). Informed consent
was obtained from all subjects, and the studies were
conducted in accordance with the tenets of the Declaration of Helsinki.
The video signal was digitized (512X512 pixels, 8bit gray level depth) with two frame grabbers (Matrox
PIP 1024, Dorval, Canada) and was stored in a desktop
computer. The digitized images were transferred to a
digital image workstation and were evaluated interactively with NIH Image 1.58 software (National Institutes of Health, Bethesda, MD). The imaging system
was calibrated with a millimeter grid, and the pixelto-millimeter ratio was determined as 21.5 pixels/mm,
left pupil margin
right pupil margin
METHODS
Experimental
In the design of the experiments, care was taken to
minimize nonsystematic errors for analysis of systematic differences between different measurement conditions and evaluation procedures. Therefore, a head
and chin rest, similar to that used for slit lamp examinations, was used to control head position and distance to the recording apparatus. Subjects were instructed to keep their heads still throughout the measurements in all gaze directions. However, the head
was not completely immobilized (for instance, with a
bite bar or a strap). All values measured or calculated
in degrees (°) were converted into prism diopters
(PD) by the equation PD = 100 • tan°.
The apparatus (Fig. 2) consisted of a horizontal
fixation bar located 50 cm from the eyes. Fixation
targets were positioned at cyclopean gaze angles of
±18.8° (34.04 PD), ±10° (17.63 PD), ±5° (8.75 PD),
and 0° (0 PD). Two charge-coupled device cameras
(SDT 3440, Seitner Datentechnik, Seefeld, Germany),
one for each eye, equipped with extension tubes and
lenses, were used for imaging the anterior eye segments at a 50-cm distance. A photo flash unit (Tumax
A16) mounted 10 cm vertically under the central 0°
left limbus
right limbus
FIGURE 3. Horizontal positions P of ocular structures involved in the measurement of the horizontal corneal reflex
decentrations relative to the limbus center (xiimbus) and to
the entrance pupil center (*pUpj|). The horizontal entrance
pupil center position (centerpiipi|) was calculated as [P(right
pupil margin) + P(left pupil margin)]/2. The horizontal
limbus center position (centeriimb) was calculated as [P(right
limbus) + P(left limbus) ]/2. Consequently, in the right eye,
*puPii = P(corneal reflex) — center pil|jill and x\;in\JUS = P(corneal reflex) — center| imblis . In the left eye, ,\pUpj| = — [P(corneal reflex) —centerpupil], and x\imbUi = — [P(corneal reflex)
—centeriimbus,}. These definitions were chosen to have a positive sign of x for a nasal shift of the corneal reflex, consistent
with a corresponding positive angle K, and vice versa for a
temporal shift.
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Investigative Ophthalmology & Visual Science, November 1997, Vol. 38, No. 12
with no significant bias between central and peripheral portions of the image.
The examiner took the x,y coordinates of the lateral limbus margins, the entrance pupil margins, and
the corneal reflex in each eye. As illustrated in Figure
3, the quantities calculated from this data were the
horizontal limbus diameter (LD), the positions of the
limbus center and the entrance pupil center, and the
horizontal distance of the corneal reflex relative to the
entrance pupil center (xpupil) and the limbus center
(limbus) • A nasal decentration x carried a positive sign,
and a temporal decentration x carried a negative sign.
Three examiners participated in the evaluation of
the image data. There were six subjects with light irides and four subjects with dark irides. On average,
the localization of the horizontal position of the nasal
and temporal limbus appeared subjectively a little
more difficult than the localization of the nasal and
temporal entrance pupil margins, because of the presence of a gray-level transition zone from the iris to
the sclera that represented the position of the limbus,
whereas there was a sharp contrast between entrance
pupil and iris, unless the subject had a dark iris.
Error levels associated with image evaluation were
assessed in preliminary trials, and the typical discrepancy in the measurement of the corneal reflex position between examiners and between reevaluations of
the same image by the same examiner was 0 or 1
pixels.
The overall reproducibility of the measurement
of the distance between ocular landmark and corneal
reflex in the study with all 10 subjects was assessed in
the following way: In the binocular pictures taken in
the primary position, the mean and standard deviation
of the difference of measured corneal reflex position
x, within each set of three images taken of one eye in
one gaze direction, was calculated in all 10 subjects
for entrance pupil center and limbus center, separately. The standard deviation, SD(x), of the corneal
reflex position x in a set of three images served as an
indicator of its reproducibility in each subject.
Therefore, the 20 standard deviations, SD(x), obtained for all right and left eyes and each landmark
(entrance pupil-limbus center), were averaged and
their standard deviations calculated: mean of SD(xHm.
blls) = 0.048 mm with a SD of SD(xlimbus) = 0.031 mm,
and mean of SD(Xpupii) = 0.044 mm with a SD of
SD(Xplipii) = 0.031 mm. There was no statistically significant difference between the mean values of pupiland limbus-based measurement of the corneal reflex
position. Therefore, the reproducibility of corneal reflex position was the same for pupil-based and limbusbased measurement. The slightly smaller value of the
mean of SD(xpupji), compared with SD(xumbus) is in
keeping with the examiners' subjective impressions
that the margins of the entrance pupils were easier to
locate than the somewhat less well defined limbi: In
general, entrance pupil margins had a sharp dark contrast with the iris, whereas the limbus could be described as a pixel region with a transition from one
gray level to another. In the latter instance, the examiner had to delimit the middle of the transition zone
to mark the limbus position, which appeared less well
denned than the entrance pupil margins.
The same checks were executed for secondary
gaze positions and for monocular images; and again,
no significant difference between pupil- and limbusbased measurements were observed in the reproducibility of the corneal reflex position.
The 0.05-mm magnitude of the mean reproducibility roughly equaled the pixel size; therefore, it can
be concluded that the determination of the corneal
reflex position was limited by the minimum pixel (1pixel) error in this setup, for either condition. Bear in
mind that measurement of the corneal reflex position
involved three measurements, each of which could be
affected by, for instance, variations of fixation, and by
the pixel error (error inherent to digitized images)
that could accompany the determination of the nasal
and temporal entrance pupil margins or of the nasal
and temporal limbus and of the corneal reflex position. This could have entailed the maximum pixel
error (3 pixels), but the current data indicate that the
reproducibility was far better than that.
In the workup of the patients, the keratometer
readings and their objective autorefractions (RK-1;
Canon Europe, Amstelveen, The Netherlands) were
taken. The keratometer readings were converted to
give the corneal curvature in the horizontal meridian.
Most subjects were moderately myopic (mean spherical equivalent [SE] approximately —2 diopters [D];
minimum SE -5.6 D, maximum SE +1.7 D). The
largest measured astigmatic component was 2.5 D.
Data Evaluation
In the first step of the data analysis, the measured
displacements x were plotted against the sine of the
gaze angle (5 to determine the experimental value of
/*and the offset xo(fi = 0°) compared with the relationship x = h'sin(P) + xo (derived from equation 1).
This was done for each subject, for the entrance pupil
center and the limbus center reference, and in both
eyes separately. The values of h in the 10 subjects for
each reference center were plotted against the measured corneal radius in the horizontal meridian for
right eyes and left eyes separately (Figs. 4A, 4B, 5A,
5B). Mean corneal radius in the right eyes (r = 8.06
± 0.26 mm [SD]) and in the left eyes (r = 8.07 ± 0.30
mm) were slighdy above the average adult value of r
= 7.8 mm, in keeping with the slight predominance
of myopic subjects with larger globes participating in
the study. From this data, a mean hpUpi\ = 4.884 mm
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Accuracy of Ocular Alignment Measurement
right eyes, limbus center
CO
X)
A
7.4
7.6
7.8
8
8.2
8.4
8.6
corneal radius r / mm
left eyes, limbus center
5.8
and hpUp\\ (r = 7.8 mm) = 4.854 mm, with HRpupii (r
= 7.8 mm) = 11.89%nm (21.05 PD/mm), with both
HRs falling in the range of reported HRs. 612 " 151819
The calculated eye rotation angles ?7REpupii and r)Xs.
pupii (where RE = right eye and LE = left eye) were
computed, using HRpupil mean, and equation 2. In the
same manner, ?7RE Umbus and ?7LE umbus were computed
using HRiimbus mean. Two ways of interpreting eye
alignment data may be used clinically to measure ocular misalignment. These two options were simulated 30
with our data.
Condition 1, binocular approach: The angle of strabismus was obtained by subtracting the value of 77 (/?)
for any gaze direction ft from the value 77 (/5 = 0°) of
the fellow eye in the primary position. This should be
a good estimate of the manifest ocular misalignment
under binocular viewing conditions. However, differences in the offsets XQ of the corneal reflex in the
primary position in the right and left eye could lead
right eyes, pupil center
7.6
7.8
8
8.2
corneal radius r / mm
FIGURE 4. Plots of the distance humbu:i (obtained from linear
fits of measured Xiimbus versus sin[/5]) against the horizontal
corneal radius r (A) in the right eye, and (B) in the left eye,
for the limbus center condition.
(right eye: 4.840 ± 0.320 mm; left eye: 4.927 ± 0.347
mm) and a mean hyimhus = 5.243 mm (right eye: 5.233
± 0.290 mm; left eye: 5.253 ± 0.308 mm) were derived, and linear fits were obtained, one for right and
one for left eyes.
The HRs were calculated as (180°/TT)'arcsin (1
mtn/A)/l mm, as explained above. The differences
of the mean HRpupii and HR|imbus were found to be
statistically significant for right eyes and left eyes, separately. After averaging between right and left eyes, we
found HR p u p i l mean = 11.82°/mm (20.92 PD/mm),
and HR, imbus mean = l T / m m (19.43 PD/mm). These
HRs were used in the rest of the study to calculate 77
from observed corneal reflex positions.
For comparison, the linear fits of Figures 4A, 4B,
5A, and 5B were used to calculate the HR for the
average corneal radius in adults, r = 7.8 mm. This
resulted in hVimbus (r = 7.8 mm) = 5.165 mm, and
HR,imbus ( r = 7.8 mm) = 11.16°/mm (19.73 PD/mm),
7.4
7.6
7.8
8
8.2
8.4
8.6
8.4
8.6
corneal radius r / mm
left eyes, pupil center
B
7.4
7.6
7.8
8
8.2
corneal radius r / mm
FIGURE 5. Plots of die distance hpupu (obtained from linear
fits of measured Xplipii versus sin[/?]) against the horizontal
corneal radius r (A) in the right eye, and (B) in the left eye,
for the entrance pupil center condition.
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Investigative Ophthalmology & Visual Science, November 1997, Vol. 38, No. 12
to errors in this approach. The differences
nocC/3) = !7REG0) " ViAP = 0°), Ai7LEbinoc(/3) =
- rjRZ(P = 0°), with 0 = ±5° (8.75 PD), ±10° (17.63
PD), and ±18.8° (34,04 PD) were used to simulate
angles of strabismus of the same size. They were computed in the right and left eyes of the 10 subjects,
together with their mean values and SDs. These followed a normal distribution, and the confidence intervals for the measured angle of strabismus were calculated as twice the standard deviation.
Condition 2, monocular approach: The angle of
strabismus was obtained by subtracting the value of
rj(P) for any gaze angle 0 from the value r)(0 = 0°)
of the same eye in the primary position. This equals,
or at least comes close to, the orthoptic simultaneous
prism and cover test procedure.31 The differences,
AT)nKMIOC(0) = v(P) ~ l(/3 = 0°), with 0 = ±5° (8.75
PD), ±10° (17.63 PD), and ±18.8° (34.04 PD), should
equal angles of strabismus of the same size. The advantage of this approach is that the offset #o of the corneal
reflex in the primary position is automatically eliminated— that is, the angle K32 or any other individual
offset does not bias the measurement. The simulated
angles of strabismus A?7monoc(/3) were computed in the
right and left eyes of the 10 subjects with their mean
values and SDs. Using the same procedure as for condition 1, assessment was made to determine whether
the simulated angles of strabismus could be treated
as normally distributed. The data were found to be
normally distributed for bodi conditions, and the confidence intervals for the differences between calculated and expected angles of strabismus were calculated as the mean difference plus twice the SDs33 to
assess the error levels.
Finally, to illustrate the variability of the position
of the entrance pupil center relative to the limbal
center, the horizontal distances of the entrance pupil
center to the limbus center, centerpupil-centeriil1lbus,
were plotted against the sine of the gaze angle /3 for
all 10 subjects in the right and left eyes.
RESULTS
Four twin box plots (Figs. 6A, 6B 7A, 7B, 8A, 8B, 9A,
9B) allow for a comparison of accuracy of misalignment measurement with entrance pupil and limbus
center reference. In the box plots, the ordinate indicates the difference between measured and expected
angle of strabismus: Boxes show 25% and 75% confidence intervals, and T bars indicate 95% confidence
intervals; data points outside the 95% confidence interval are shown separately. The boxes represent the
seven simulated angles of strabismus, from right
(-18.8°, -34.04 PD) to left (+18.8°, +34.04 PD). Note
that the central 0° box refers to the differences of
measured angles in the primary position, below which
condition [\) - iimbus, ngnt e y e s
condition (1) - limbus, left eyes
FIGURES 6 TO 9. Box plots (for Figures 6 to 9) of differences
between calculated and expected angles of strabismus for
binocular condition 1 and monocular condition 2. Central bar
indicates median, upper and lower box borders refer to 25%
to 75% confidence interval limits, and T-bars refer to 95%
confidence intervals. Data points outside these limits are
shown as individual points.
FIGURE 6, Condition 1: box plots of the difference between
calculated eye rotation angles i7iilllbus and 77iiinblis (/? = 0°) of
the fellow eye, (A) right eye misaligned and (B) left eye
misaligned, for six simulated angles of strabismus calculated
from limbus center reference data.
a manifest strabismus may not be distinguished from
alignment data occurring in orthotropic subjects. In
other words, 0° data delimit the threshold for detection of a manifest misalignment in the primary position.30'34'35
Condition 1 simulated binocular angles of strabismus: Figures 6A and 6B show box plots of the difference between expected and simulated angles of strabismus (limbus center), for (A) right eyes and for (B)
left eyes. Figures 7A and 7B show the box plots of the
difference between expected and simulated angles of
strabismus (entrance pupil center), for (A) right eyes
and for (B) left eyes. In the primary position, the
maximum 95% confidence intervals for right and left
eyes were limbus, ±3.174° (5.5 PD), and entrance pupil, ±5.217° (9.1 PD). Clearly, errors were smaller for
the limbus option, for all simulated angles of strabismus. The 95% confidence levels at ±18.8° of simulated angle of strabismus were, at most, 26.4% (limbus) and 37.0% (entrance pupil) of the amount of
the simulated angles of strabismus.
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Accuracy of Ocular Alignment Measurement
condition (1) - pupil, right eyes
condition (1) - pupil, left eyes
B
FIGURE 7. Condition ]: box plots of the difference between
calculated eye rotation angles 77pupi| and 77piipi| {0 = 0°) of
the fellow eye, (A) right eye misaligned and (B) left eye
misaligned, for six simulated angles of strabismus calculated
from entrance pupil center reference data.
Condition 2 simulated monocular angles of strabismus: Figures 8A and 8B show box plots of the difference between expected and simulated angles of strabismus (Hrnbus center), for (A) right eyes and for (B)
left eyes. Figures 9A and 9B show the box plots of the
difference between expected and simulated angles of
strabismus (entrance pupil center), for (A) right eyes
and for (B) left eyes. In the primary position, the
maximum 95% confidence intervals for right and left
eyes were limbus, ±1.07° (1.9 PD), and entrance pupil,
±1.3° (2.3 PD). Errors were slightly smaller for the
limbus option for all simulated angles of strabismus.
The 95% confidence levels were, at most, 19.3% (limbus) and 21.9% (entrance pupil) of the amount of
the simulated angles of strabismus.
It could be speculated that with a higher camera
resolution, systematic differences between pupil- and
limbus-based measurement reproducibility could have
been noted, if they were present. However, such error
levels would, on average, have had to be smaller than
or equal to at least 1 pixel. Consequendy, any systematic difference between pupil-and limbus-based mean
corneal reflex distances that was markedly larger than
1 pixel must be caused by genuine differences between these conditions, in that the image evaluation
was done in the same images.
Figure 10 (right eyes only) reveals that the en-
2603
trance pupil and limbus center positions, plotted
against the sine of the gaze angle, varied for different
gaze angles. In most subjects, the differences decreased when looking to the left (negative sines). For
some of the plots, satisfactory linear regressions could
be obtained. At approximately an 11° (19.43 PD) gaze
angle to the left, entrance pupil and limbus centers
of the right eyes coincided, on average, within a range
of ~±0.25 mm. The average distance between entrance pupil center and limbus center in the primary
position is indicated by the differences at sin(0°): On
average, the entrance pupil centers were decentered
~0.18 to 0.23 mm nasally, in keeping with independent data.28 The difference between both increased
with a more temporal position of the eye, because the
projected distances between the entrance pupil and
limbus centers appeared larger when viewed
obliquely, in that the entrance pupil plane is slightly
behind the limbus plane (Figs. 4, 5). The data for left
eyes (not shown in Fig. 10) suggest similar conclusions. Note, however, that in one subject (top plot,
black triangles), the difference was nearly constant
and linearity was limited in some of the plots. This
could be because of the entrance pupil movement;
the limbus movement should be linear with the sine
of the gaze angle, in keeping with the eye model in
Figure 1. However, it could be speculated that in some
condition (2) - limbus, right eyes
condition (2) - limbus, left eyes
FIGURE 8. Condition 2: box plots of the difference between
calculated eye rotation angles rj|imbus and 77timl)lls (0 = 0°) of
the same eye, (A) right eye misaligned and (B) left eye
misaligned, for six simulated angles of strabismus calculated
from limbus center reference data.
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Investigative Ophthalmology 8c Visual Science, November 1997, Vol. 38, No. 12
condition (2) - pupil, right eyes
condition (2) - pupil, left eyes
FIGURE 9. Condition 2: box plots of the difference between
calculated eye rotation angles i7pupil and J7pupi, (/3 = 0°) of
the same eye, (A) right eye misaligned and (B) left eye
misaligned, for six simulated angles of strabismus calculated
from entrance pupil center reference data.
ter is a much less reliable landmark than the limbus
center, when used to measure the relative position of
the corneal reflex. Unpredictable shifts of the entrance pupil center29 and variable anterior chamber
depths3 contribute to this unsystematic error.
To examine whether varying pupil size could have
been the reason for the lower precision of eye alignment measurement, horizontal entrance pupil and
limbus diameter variations were analyzed in the same
way as the reproducibility of corneal reflex position
(see Methods): For each eye, the mean diameter of
the limbus and of the entrance pupil, and its standard
deviation, SD (diameter), were determined from a triplet of images in the primary position. Then the mean
SD (diameter) and standard deviation of SD (diameter) of the 20 individual standard deviations of the
diameters were computed. Mean standard deviations
indicated a significant difference (P < 0.0001, paired
Student's Rest) between entrance pupil and limbus
diameter variation: Mean SD(pupil diameter) = 0.456
mm, with a standard deviation of SD (pupil diameter)
= 0.154 mm, mean SD (limbus diameter) = 0.132 mm,
with a standard deviation of SD (pupil diameter) =
0.137 mm. Note that the mean SD(pupil diameter)
was much larger than the pixel error. This indicated
that the reproducibility of diameter measurements
was equal for the entrance pupil and for the limbus.
of the cases, the limbus itself was seen through the
peripheral portions of the cornea, and hence could
be treated as an entrance limbus itself. The movement
of its center would then be more difficult to predict.
DISCUSSION
Brodie et al3'6 favored the use of the limbus as a landmark, partly for practical reasons, pardy because they
suspected that this choice would reduce error levels
caused by varying depths of the anterior chamber.3 In
the same study, a nominal value for h was computed,
but an experimental dependence on the corneal radius was not established, and a single HR was advocated regardless of pupil- or limbus-based measurement, confirming previous similar reasoning.19
In the eye model shown in Figure 1, the difference
between the entrance pupil center and the limbus
center as a landmark and the resulting behavior may
be explained readily: The entrance pupil center is, on
average, ~0.4 mm axially behind the limbus center,
and is nasally shifted by approximately 0.25 mm. This
can be adjusted for by using appropriate HRs.
However, there is some variation between right
and left eyes, and the scatter of the difference is sometimes large. This suggests that the entrance pupil cen-
-.3
-.3
-.2
-.1
0
.1
sin (gaze angle)
FIGURE 10. Plot of the horizontal distance between entrance
pupil center and limbus center, centerpiipii — centerijlllbll5!
versus sin(/?) in the right eyes. In this plot, a linear behavior
points to a constant ratio of hpUpn to hy,mhm. The slopes of
the graphs are proportional to individual ratios of h. The
differences change approximately linearly with sin(/3), but
the positions and slopes of the curves vary significantly between individuals, and in some instances, nonlinear behavior is evident.
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Accuracy of Ocular Alignment Measurement
The variability of the entrance pupil diameter, however, was threefold, compared with the limbus diameter variability and compared with the measurement
reproducibility of the diameters. Similar results were
obtained for the other gaze positions.
This means that the most probable explanation
for the difference of precision between limbus and
entrance pupil measurement of eye alignment in binocular images is given by the larger variations of entrance pupil size, and hence by the associated shifts
of the entrance pupil center without an ocular rotation. By contrast, at least in our model (Fig. 1), the
limbus center cannot shift in a similar way, because
the limbus is a rigid structure. However, the simple
anatomic model shown in Figure 1 does not represent
all of the real effects that could be expected: Coincidentally, the effects of neglected corneal asphericity
and the linearization of the sine in equation 1 compensate each other to some extent, such that the validity of equation 2 is extended to larger angles. Exact
ray tracing analysis with software introduced elsewhere38 and modified to model aspheric corneas,
showed that when a spherical shape was taken into
account, the limbus must be assumed to be a structure
imaged through the peripheral portions of the cornea, lying ~0.5 mm behind the corneal surface. This
means that the visible limbus is an "entrance limbus"
analogous to the entrance pupil.
The study's maximum error in eye alignment measurement is tangible in the binocular approach (Figs.
6A, 6B, 7A, 7B): Here, the errors of entrance pupil
data became approximately twice as large as those of
limbus data. This is of considerable clinical relevance,
because most measurements of eye alignment are recorded relative to the entrance pupil center as binocular difference of ocular alignment between right and
left eyes, in that this does not require pictures with
one eye covered.
In the monocular approach (Figs. 8A, 8B, 9A, 9B),
any offset in terms of entrance pupil center shift, angle
K or other parameter, whatever its cause, was cancelled, because only the difference of the angle r) between two gaze directions was computed. This reduced the error levels such that differences between
entrance pupil and limbus center measurements became small, if not negligible. However, this measurement procedure, which requires covering the right
eye and the left eye, often cannot be applied in infants,
toddlers, and preschool children, in whom screening
with an objective ocular alignment technique is most
useful.
It is of clinical and practical interest to note that
when coaxial infrared lighting screening techniques24'25 are used, the entrance pupil margins are
easily identifiable structures with simple thresholding
algorithms in digitized images. This is because the
2605
dominant image portions are the fundus reflexes
against which the corneal reflex appears as a single
bright reflex and the entrance pupil margins as sharp
boundaries of the fundus reflex. The limbal region is
imaged with less contrast under these lighting conditions, which may lead to neglect of the limbus center
as a landmark. This explains, possibly together with a
limited resolution of the CCD camera used in the
apparatus, the limited accuracy of 8 to 10 PD (4.6° to
5.7°) for detection of ocular misalignment reported26
elsewhere. In the current study, despite subjective impression, the limbus could be located with precision
interactively. We also showed elsewhere that image
processing may be used to automate the detection of
limbus structures in digitized images36 and with nearly
the same precision as is possible in detection of the
entrance pupil margins.
These results prove that the accuracy of measurement of ocular misalignment depends decisively on
the choice of the reference landmark. Different HRs
should be used, depending on this choice, because
the mean limbus value was 1.5 PD/mm lower than
the entrance pupil value in this study. Normative values for h, the distance that, together with the corneal
radius and limbus diameter, governs the HR, were
established for limbus and entrance pupil center landmarks for an average human adult eye with corneal
radius r= 7.8 mm. Most data in the literature 512 " 141819
refer to the entrance pupil center only or do not take
into account systematic differences between pupiland limbus-based measurement. For specific populations, in which the corneal radius may be measured,
the HR may be determined from our data by interpolation or extrapolation of h, unless the HR is calculated
through plots of measured corneal reflex shift versus
gaze angle. Using this alternative for evaluation of the
present alignment data led to the same systematic difference between pupil- and limbus-based HRs. We
conclude that, as a rule, limbus-based HRs are 1 to 2
PD/mm lower in humans than are pupil-based HRs.
In animals, similar systematic effects can be predicted.
After the current study, which was conducted
solely on control subjects, to minimize errors that
would impede the accurate assessment of misalignment measurements, a comparison of methods could
be done in strabismic subjects with orthoptic control
data. However, care must be taken to avoid subjectrelated, time-dependent fluctuations of the angle of
strabismus when different techniques are compared.
Because random errors would be larger than those in
this study, the number of strabismic subjects would
have to be much higher to ascertain the same amount
of systematic difference. Because of the design of the
current study, it was possible to examine differences
between techniques with a relatively low number of
subjects.
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Investigative Ophthalmology & Visual Science, November 1997, Vol. 38, No. 12
Other studies concluded that h was usually independent of age after 3 years of age and that the statistical correlation of the HR with the corneal radius was
too weak19 to be relevant. Although this may be true,
the systematic yet small difference in HR related to the
choice of the landmark should be taken into account,
because it would improve reported insufficient thresholds for ocular misalignment detection. 26
The highest accuracy of ocular alignment measurement was obtained with the monocular approach,
in which the choice of limbus or entrance pupil center
did not make a difference. However, such a measurement strategy depends on the cooperation of the subject.
In conclusion, we recommend the use of the limbus center for screening and measuring applications
with corneal reflexes, whenever recorded images are
used. The threshold for detection of ocular misalignment in the binocular approach, the 95% confidence
interval, was 3.174° (5.5 PD), compared with nearly
twice that error in the widely used entrance pupil center evaluation. To increase the precision of ocular
alignment measurement further, measurement with
corneal and posterior crystalline lens reflexes from
three light sources30'34'35'37 could be used. Using this
methodology and the alignment data of this study, the
threshold for detection of ocular misalignment was
2.1° (3.7 PD).
If the corneal reflex position is simply estimated
during a clinical examination, without recording, we
speculate that the limbus option probably does not
improve the accuracy of the eye alignment assessment:
The errors in the estimation of the asymmetry of the
corneal reflex position, especially regarding the limbus center, are probably much larger than the potential increase in accuracy indicated by the results of the
current study.
A similar effect can be expected if the spatial resolution of the recorded images is low—that is, on the
same order as the inherent errors of binocular corneal
reflex position measurement: If the limbus-center-related threshold in the primary gaze is taken from the
current data as 3.174° (5.5 PD), and a typical HR is
~21 PD/mm, then a critical limit must be expected
for a resolution of ~0.25 mm/pixel, or 4 pixels per
millimeter as a hardware magnification factor (compared with 21.5 pixels per millimeter in this study,
with one camera per eye). To image spatial differences
in the corneal reflex position of 0.25 mm reliably, the
required pixel density must be higher than this limit,
depending on the type of hardware used.39
Fortunately, 35-mm photographs or slides have
much better resolution than 0.25 mm, 39 and most of
the currently available low-cost analog CCD cameras
and digitizing frame-grabbers frequently used for eye
alignment measurement perform well enough to sur-
pass the critical value of 4 pixels per millimeter; some
perform well even if only one camera is used to image
both eyes at a time.
However, care must be taken not to overestimate
the spatial and temporal resolution that can realistically be expected with some CCD cameras: For instance, in the interlace mode, 39 half-images are typically read out at 50 to 60 Hz, which can cause motion
artifacts (shifted half-images of moving objects), especially in eye-tracking applications; and asynchronous
pixel readout or aliasing may cause errors of at least
1 pixel width.39 The maximum spatial resolution is
also limited by the amount of CCD sencor elements
per row and number of rows, signal-to-noise ratio,
bandwidth of the analog video signal, and the frame
grabber resolution. Such effects limit the benefit of
any subpixel evaluation technique. 16 These problems
may be avoided by using appropriate imaging devices—for instance, analog full-frame-transfer CCD
cameras and cameras with a synchronized pixel transfer, or CCD cameras with a digital output, which do
not require a frame grabber.
Therefore, in addition to the proper choice of the
landmark for corneal reflex position, we recommend
that the design of recording devices for screening,
clinical, and research purposes be guided by the desired measuring accuracy or threshold for detection
and that the hardware be carefully selected and tested
to ensure that those criteria are met.
Key Words
accuracy, angle of strabismus, corneal reflex, instrumentation, paraxial ray tracing analysis
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