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From the Section of Neurobiology of the Eye, Institute for Ophthalmic Research, Tübingen, Germany.
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Abstract |
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 Top Abstract MethodsResultsDiscussionAppendix 1References |
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METHODS. The first, third, and fourth Purkinje images (P1, P3, P4) were visualized using a single infrared (IR) light-emitting diode (LED), a planar lens (F = 85 mm; f/number of 1.4), and an infrared sensitive analog video camera. Software was developed to mark pupil edges and positions of P1, P4, and P3 with the cursor of the computer mouse, for three different gaze positions, and an automated regression analysis determined the gaze position that superimposed the third and fourth Purkinje images, the gaze direction for which the lens was oriented perpendicularly to the axis of the IR LED. In this position, lens decentration was determined as the linear distance of the superimposed P3/P4 positions from the pupil center. Contrary to previous approaches, a short initial fixation of a green LED with known angular position calibrated the device as a gaze tracker, and no further positional information was necessary on fixation targets.
RESULTS. Horizontal and vertical kappa, horizontal and vertical lens tilt, and vertical lens decentration were highly correlated in both eyes of the subjects, whereas horizontal decentration of the lens was not. There was a large variability of kappa (average horizontal kappa –1.63° ± 1.77° [left eyes] and +2.07° ± 2.68° [right eyes]; average vertical kappa +2.52° ± 1.30° [left eyes] and +2.77° ± 1.65° [right eyes]). Standard deviation from three repeated measurements ranged from 0.28° to 0.51° for kappa, 0.36° to 0.91° for horizontal lens tilt, and 0.36° to 0.48° for vertical lens tilt. Decentration was measured with standard deviations ranging from 0.02 mm to 0.05 mm. All lenses were found tilted to the temporal side with respect to the fixation axis (on average by 4.6°). They were also decentered downward with respect to the pupil center by approximately 0.3 mm.
CONCLUSIONS. Lens tilts and positions could be conveniently measured with the described portable device, a video camera with a large lens. That the lenses were tilted to the temporal side in both eyes, even if corrected for kappa, was unexpected. That they were displaced downward with respect to the pupil center could be related to gravity.
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Methods |
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TopAbstractMethodsResultsDiscussionAppendix 1References |
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Setup
The setup consisted of a CCIR monochrome infrared-sensitive video camera (DMK 3002-IR/C; available through TheImagingSource at www.theimagingsource.com), and a planar lens (F = 85 mm; f/1.4; #7129754; Zeiss, Oberkochen, Germany) attached to the camera by a 30-mm extension ring (Figs. 1A 1B , left). This arrangement resulted in an image magnification in the video image of 47.2 pixel/mm at a distance of 255 mm from the front surface of the lens. An analog charge-coupled device camera had to be used because none of the tested digital cameras achieved sufficient infrared sensitivity to visualize the third Purkinje image (P3) with good contrast. The video signal was fed to a computer or laptop with either an analog-to-fire wire converter box (DFG/1394 to 1e) or an analog-to-USB converter box (DFG/USB2-lt), both available at the TheImagingSource (see above). A high-quality lens was necessary because P3 was out of focus when the first (P1) and the fourth (P4) Purkinje images were at best focus. However, when the lens aperture was stopped down to f/5.6 or f/8, the depth of focus became sufficient to visualize all three Purkinje images for simultaneous measurements (Fig. 1B , right). P3 was also initially difficult to find because it moved very quickly with changing gaze direction and disappeared behind the iris when the fixation was only a few degrees away from the camera.
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Measurement Procedure
No pupil dilatation was necessary, and the setup could measure the smallest pupil (approximately 2.5 mm) without problem. The subject’s eye had to be positioned 255 mm from the front surface of the camera lens. Given that image magnification was an important variable, the distance from the eye to the camera was controlled by a small depth of focus of only a few millimeters. Furthermore, focus was coded as sound with variable frequency, simply by multiplying the number of bright pixels in the first Purkinje image by 40 and transmitting this frequency to the speaker of the computer. At best focus, the lowest frequency of approximately 2000 Hz was emitted. To further reduce the variability of lateral eye positions, a chin rest was used. Because the gaze tracker used the position of the first Purkinje image relative to the pupil center, rather than the absolute position of the pupil center in the video frame, it had little sensitivity to head movements in the field of the camera.
Step 1.
The subject was asked to fixate the green LED (Fig. 1A) . For this gaze position, the three Purkinje images were visible in all the subjects. The user stopped frame grabbing by pressing the space bar. The edge of the pupil had to be marked at four arbitrary positions with the computer mouse, and a circle fit of the pupil margin was automatically performed. The user then had to mark the centers of P1, P4, and P3. These positions were stored by the program. The distance of P1 from the pupil center in horizontal (x) and vertical (y) direction immediately provided the angle kappa in both x and y directions. Kappa here is defined by the angular distance from the fixation axis to the pupil axis, which, in turn, is the angular eye position relative to the camera, where the first Purkinje image is centered in the pupil.7 8 That the position of P1 may also vary with the position of the eye in the video frame was not further considered because the potential error was less than the SD of the measurements (see the section on error analysis in the Discussion).
Unlike previously published procedures,1 2 the angular positions of the fixation targets, other than the green fixation LED in the beginning, had not to be known because the knowledge of kappa made it possible to track the fixation axis continuously. A Hirschberg ratio of 12°/mm was assumed in all eyes (the rotation of the eye in degrees that was necessary to move P1 by 1 mm7 9 10 ). It was previously shown that the Hirschberg ratio is largely constant over ±45° of fixation range.9
Step 2.
A second frame had to be grabbed for a different gaze position. Appropriate gaze positions had to be chosen by the user when all three Purkinje images were visible in a new position. To stimulate changes in gaze position, the subject was asked to read numbers printed on the plastic ring that was attached to the camera lens and that also carried the LEDs (Fig. 1B , left). Again, the pupil edges and P1, P4, and P3 had to be marked.
Step 3.
A third frame had to be grabbed with yet another position of gaze. Once the information on pupil and Purkinje image position was recorded, the program performed regression analysis for the distance of P3 and P4 in x-direction and y-direction versus the angular direction of the fixation axis in x- and y-direction. The regression lines were immediately displayed on the screen (Fig. 1C , top right). If the regression did not achieve significance (correlation coefficient R < 0.95), the program provided an error message and a new set of data had to be taken. However, this happened in only a few cases (less than 5%).
The intersections of the regression lines with the abscissae provided the direction of gaze that had to be taken to superimpose P3 and P4, a gaze position where the crystalline lens was oriented exactly perpendicular to the axis connecting the green fixation LED to the pupil center. Lens tilt in horizontal and vertical directions, relative to the fixation axis, could be deducted from the negative value of the respective gaze angles.
Lens decentration was measured relative to the pupil center because it was not possible to determine the exact position of the chief ray of the fixation axis in the pupil. The position of the pupil center was already known. The position of the lens center was determined in the gaze position where P3 and P4 were on top of each other. The lens decentration was then given by the linear distance of the superimposed P3 and P4 from the pupil center (given in mm).
No ray tracing was performed, unlike several previous studies.3 4 11 It was concluded that the presented measurement algorithms were sufficient within the range of measurement noise (see error analysis in the Discussion) and that more detailed analyses of the optical parameters would not improve the quality of the data (also schematic eyes have standard deviations).
Signs were important for correct interpretation of the measured numbers. Sign definitions are shown in Figure 2 .
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Results |
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TopAbstractMethodsResultsDiscussionAppendix 1References |
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Discussion |
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TopAbstractMethodsResultsDiscussionAppendix 1References |
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Furthermore, the optics of the current system are simple. No collimated light sources and no telecentric lenses were used. Magnifications of P3 and P4 were not very important for lens tilt measurements because only the gaze position had to be found where they were superimposed. To measure decentration, image magnification was controlled by the low depth of focus of the system and by the sound, which was modulated based on the focus.
Although the measurements were easy to perform and seemed to produce reliable data (as suggested by the standard deviations and the high interocular correlations), potential error sources have to be analyzed.
Variability and Potential Sources of Error
A number of potential error sources must be considered. First, inspection of the error bars, calculated from three repeated measurements (Figs. 3 4) , shows that they can become large. This could be explained by the observer apparently not precisely judging the positions of the centers of the Purkinje images. The problem seemed particularly relevant for P3, which often looked diffuse (Fig. 1B , right). On the other hand, P3 moved far even for small changes in the direction of gaze, denoting a small Hirschberg ratio (approximately 2.5; estimated from its movement in the pupil by approximately 4 mm, for a change in gaze direction of 10°). Therefore, even if the center of P3 was marked with an error of 20 pixels (equivalent to 0.24 mm), an error in the angle measurement of only 0.4° would have resulted. The three individual measurements were completed in approximately 1 minute. It is clear that, with more repetitions, outliers could have been identified and excluded. However, it was considered important that the device was tested in the laboratory under conditions similar to those in the clinics and that the error bars reflect the ones to be expected under such conditions.
Second, variations in eye position in the video frame could be another confounding factor. The positions of the Purkinje images move in the pupil with any lateral change in eye position. However, because the video frame size measured only 752 pixels (equivalent to 15.9 mm) and the pupil was centered in the frame with an estimated variability of ±2 mm, the change in angle was only arctan(2/255) = 0.45°. Again, this value is in the range of the measurement variability. Third, for the measurements of lens decentration, effects of magnification of P3 and P4 by the cornea were not corrected. It is known that the cornea magnifies the pupil and the distance of the superimposed P3/P4 from the pupil center by approximately 10%. This indicated that lens decentration was slightly overestimated by up to 10%, an amount of 0.03 mm for the largest decentration of 0.3 mm found in this study. Although this would be a systematic error, it is small compared with the interindividual variability and the standard deviations (Fig. 4B) .
Fourth, the measurement procedure relied on only three measurements at different gaze positions, the minimum number of points for a regression analysis. The tradeoff here was time/effort and reliability. At least, there was the possibility to evaluate the significance levels of the regression analysis online. To achieve significant regression with only three points, the data points must be very well aligned and R >0.95, providing a criterion for valid measurements. Fifth, another source of variability and potential error could be the quality of the initial fixation of the green LED, which was necessary to determine kappa. Because the fixation was binocular (the nonmeasured fellow eye was not occluded), it was possible that the fixation was better for the dominant eye and that the measurement of kappa was more reliable. This factor has not been further evaluated, but at least kappa was, on average, not significantly different in both eyes, and lens tilts were highly symmetrical in both eyes (Figs. 3C 3D) .
Sixth, Hirschberg ratios were not determined individually; rather, a literature value of 12°/mm was used.7 The standard deviations in measurements of several subject were previously found to be 1.2° to 1.5°7 (equivalent to 10%). Given that the measured gaze positions were all not very far out in the periphery, an error of 10% for 3° away from the pupil axis would amount to 0.3°. This is also reflected in the output provided by the gaze tracker (Fig. 1C) , which showed that the letter targets attached to the camera were properly fixated. The potential error is in the range of the standard deviations from three repeated measurements: 0.28° to 0.51° for kappas, 0.36° to 0.91° for horizontal lens tilt, and 0.36° to 0.48° for vertical lens tilt. In the interest of rapid measurements in elderly clinic patients with pseudophakic eyes who may not be able to offer a high degree of cooperation, this small uncertainty may be acceptable.
Seventh, the effects of phorias are potentially confounding. Subjects viewed the fixation targets binocularly, and the risk was that fixation was optimal only with the dominant eye. If the lens data were collected in the dominant eye, this should not be problem, but if the fellow eye was measured, it could have resulted in deviated measurement values. On the other hand, the only critical step here was the initial fixation of the green LED, used to measure kappa. To ensure proper fixation, the program accepted gaze data only if the angular standard deviation from 25 measurements was less than 0.3°. Because it is difficult to keep eye position stable without fixating a target, selecting for small standard deviations may be a valid procedure. For the other two measurements, fixation did not matter because the gaze tracker was then calibrated. The position of the fixation axis was continuously recorded. An example of the gaze tracker output is shown in Figure 1C . There was no indication that the fixation axis was significantly off when the subject was asked later to look at the green LED. Finally, the high correlation between the kappas in both eyes made it unlikely that there were consistent errors in the initial fixation task. Nevertheless, potential minor effects cannot be excluded. In future studies, it may be better to measure subjects under monocular conditions, with the fellow eye occluded.
Finally, to evaluate the repeatability of measurements in human subjects over time, the same measurements were performed in the 11 subjects at a 6-week intervals with two different copies of the setup (Fig. 6) . These measurements were highly reproducible over a period of 6 weeks and with two different copies of the device. An important potential interpretation error could the sign conventions of the measured values of kappa, lens tilt, and decentration. Detailed analysis of the signs is presented in the Appendix.
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Tabanero et al.2 provide a technical description of their device with detailed information on the calibration procedures and error sources but with data only from one eye of two healthy subjects with phakic eyes; these were in the range of the current measurements.
Kirschkamp et al.3 also do not provide binocular data for comparison, but they calculate lens tilt to the temporal by 0.2° ± 0.8° and decentration by 0.1 ± 0.1 mm to the nasal side, less than what was found in the present study. These authors also state that their (calculated) values should be viewed with caution.
In summary, lens tilts and positions could be conveniently measured without cycloplegia with the described portable device. The high degree of mirror symmetry for horizontal lens tilt and decentration in both eyes and the high level of repeatability of the measurements suggested that the device made valid measurements. That the lenses were tilted to the temporal side in both eyes, even if corrected for kappa, was unexpected and may be a remnant from our nonbinocular mammalian ancestors. That the lenses were decentered downward with respect to the pupil center might perhaps be related to gravity.
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Appendix 1 |
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TopAbstractMethodsResultsDiscussionAppendix 1References |
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First, kappa is determined from the distance of P1 to the pupil center in the horizontal (x) and the vertical (y) direction when the subject fixates the green LED: kappa horizontal = (x_P1 – x_pupil_center) x Hirschberg ratio/image magnification; kappa vertical = (y_P1 – y_pupil_center) x Hirschberg ratio/image magnification; (x_pupil_center and y_pupil_center are the x and y pixel coordinates in the video frame for the pupil center and x_P1 and y_P1 for P1; Hirschberg ratio = 12, image magnification = 47.2 pixel/mm).
If x_P1 is > x_pupil_center, kappa_x is positive, P1 is right of the pupil center, and the pupil axis is to the left of the green fixation LED, as seen from the subject. If y_p1 > y_pupil_center, kappa_y is positive, P1 is below the pupil center, and the pupil axis is above the green fixation LED.
Furthermore, because the green fixation LED is 2.53° below the IR LED that creates the Purkinje images, the y_P1 is higher in the video image (smaller y_coordinate). Therefore, an angle of 2.53° has to be added to kappa_vertical.
Second, by tracking the distance between P1 and the pupil center, the program tracks the position of the pupil axis. To convert the position of the pupil axis into the fixation axis, kappa has to be subtracted: x_gaze = (x_P1– x_pupil_center) – kappa_x; y_gaze = (y_P1 – y_pupil_center) – kappa_y.
The more positive x_gaze, the more is the fixation axis to the right. The more positive y_gaze, the more up is the direction of the fixation axis.
Third, by tracking the distance between P3 and P4 for different directions of the fixation axis, the position of the fixation axis can be found by linear regression for which P3 is on top of P4 (x_P3 – x_P4 = 0 and y_P3 – y_P4 = 0). If the fixation axis is in this position, the lens is oriented perpendicularly to the camera axis. In turn, the lens tilt angles are just the negative of the respective angles of the fixation axis. For instance, if the fixation axis is to the right of the camera to superimpose P3 and P4, the lens is tilted to the left by the same angular amount. If the fixation axis is above the green fixation LED to superimpose P3 and P4, the lens is tilted down by the same angular amount.
Fourth, lens decentration was calculated when P3 and P4 were on top of each other directly from their distance from the pupil center: decentration_x = (x_P3 + x_P4)/2 – x_pupil_center; decentration_y = (y_P3 + y_P4)/2 – y_pupil_center.
If decentration_x is positive, x_P3/x_P4 are right of the pupil center on the screen but left in the real eye, and the lens is decentered to the left. If decentration_y is positive, y_P3/y_P4 are below the pupil center, and the lens is decentered down.
Note
All measurements could have been completely automated, and some effort was made to achieve automatic detection of the pupil and all Purkinje images. However, these procedures were limited by the contrast of the third Purkinje image. Its brightness was sometimes only little above the pupil background, and irregularities in the tear film could cause false detections.
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Acknowledgements |
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Footnotes |
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Supported in part by Alcon (Freiburg, Germany) through the Steinbeis Transfer Centre for Biomedical Optics and Functional Testing (Tübingen, Germany).
Disclosure: F. Schaeffel, None
The publication costs of this article were defrayed in part by page charge payment. This article must therefore be marked "advertisement" in accordance with 18 U.S.C. 
1734 solely to indicate this fact.
Corresponding author: Frank Schaeffel, Section of Neurobiology of the Eye, Institute for Ophthalmic Research, Calwerstrasse 7/1, 72076 Tübingen, Germany; frank.schaeffel{at}uni-tuebingen.de.
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References |
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TopAbstractMethodsResultsDiscussionAppendix 1References |
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