Binocular Lens Tilt and Decentration Measurements in Healthy Subjects with Phakic Eyes

Abstract

purpose. Tilt and decentration of the natural crystalline lens affect optical quality of the foveal image. However, little is known about the distributions of these variables in healthy subjects with phakic eyes and about their correlations in both eyes. A simple, portable, easy-to-use, and partially automated device was developed to study lens tilt and decentration in both eyes of 11 healthy subjects with phakic eyes.

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.

Several devices have already been developed to measure the tilt of the crystalline lens and its decentration in human eyes. ^{1 2 3 4 5 6} This raises the question as to why yet another attempt was made in the present study. The reasons were that the distributions of lens tilt and decentration in healthy phakic eyes have not been well described, the correlations between these variables in both eyes have not yet been studied in detail, the previously used devices were often difficult to use and required extensive calibrations and calculations, there is a need for an easy-to-use setup for clinical use, and the devices were not very portable because the fixation targets had to be presented under defined visual angles. In this study, a setup was composed (consisting of a video camera with a large lens and a laptop computer) and programmed that provided data on lens tilt and decentration in approximately 1 minute. Because the device also worked as a gaze tracker, no positional information was necessary for fixation targets. Given that the data analysis was automated, no background knowledge on the procedure was necessary. 

Methods

Subjects

Eleven subjects from the laboratory with no known ocular abnormalities, other than myopia, were measured. The measurements took place without spectacle correction because spectacles change the Hirschberg ratio in correlation with their optical power, ⁷ and because they introduce disturbing specular reflections. The ages of the subjects ranged from 25 to 53 years. The research adhered to the tenets of the Declaration of Helsinki and informed consent was obtained from the subjects after explanation of the nature and possible consequences of the study. The device was evaluated by the department for medical technology of the University of Tübingen for potential risks of hazard, and an official report was obtained documenting its harmlessness. Furthermore, the Ethics Commission of the Medical Faculty of the University of Tübingen was consulted and approved the study. 

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. 

Two light sources were attached to the lens: an infrared LED (0254209504; 5-mm diameter; Conrad Electronics www.conrad.de), which created the Purkinje images, and a green fixation LED, attached 2.53° below the infrared (IR) LED. LEDs were positioned 39 mm and 27.5 mm from the camera axis, equivalent to visual angles of 8.7° and 6.2°, respectively. An IR-transmitting cutoff filter (B+W 49 ES 092; Schneider, Bad Kreuznach, Germany) was necessary in front of the camera lens to optimize the contrast of the Purkinje images in the pupil (Fig. 1B) . Image processing and automated data analysis were performed by software written in Visual C++ (Fig. 1Cshows the screen output of the software). 

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 mm ^{7 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 . 

Statistical Analysis

Orthogonal regression analysis, which does not assume an independent and a dependent variable, was used to study correlations between both eyes (JMP, version 5.5; SAS Institute, Cary, NC), and paired t-tests were used to test for differences in kappa between both eyes. Finally, t-tests were used to determine whether the lenses were significantly tilted relative to the fixation axis or were displaced relative to the pupil center. 

Results

Interocular Correlations of kappa, Lens Tilt, and Lens Decentration

In the horizontal direction, kappa and lens tilt showed mirror symmetry in both eyes (Figs. 3A 3C ; P < 0.01). Horizontal lens decentration was more scattered and not correlated (Fig. 1E) . With the use of orthogonal regression, the slope in the regression between kappa from the left versus the right eye was close to −1 (−0.91), indicating mirror symmetry. The ranges of variability in all the measured variables were impressive. Horizontal kappa ranged from −1.94° to +6.52°, and the lens was strikingly tilted to the temporal side, relative to the fixation axis, by −10.24° in one subject but only −1.68° in another (Fig. 3C) . In the vertical plane, kappa, lens tilt, and lens decentration were positively correlated (Figs. 3B 3D 3F) . 

Lens Tilt and Decentration in Both Eyes

Figure 4shows lens tilt (Fig. 4A)and decentration (Fig. 4B)in both horizontal and vertical directions. With respect to the fixation axis, lenses were significantly tilted “up” (the top of the lens tilted more toward the back and the bottom tilted more toward the front (P < 0.001; Fig. 4A )). Even more strikingly, all lenses were tilted temporally compared with the fixation axis, on average by approximately 4.6° (by more than 10° in one of the subjects). Decentrations were also highly significant. The lenses were displaced to the temporal side of the pupil center by approximately 0.1 mm (P < 0.01) and down by more than 0.3 mm (P < 0.001). It should be kept in mind that the measured decentration could also result from a displacement of the pupil center relative to the “fixation axis.” There is evidence that the pupil is displaced nasally by approximately 0.25 mm, ¹² which can account for part of the apparent temporal decentration of the lens. 

Illustration and Summary of Lens Tilt and Decentration Data

Figure 5provides a summary and illustrates in which directions the lens were tilted and decentered. A striking observation was that lenses were tilted to the temporal side not only relative to the fixation axis (by approximately 4.6°) but also to the pupil axis, after kappa was subtracted (tilt still approximately 3°; see data for the horizontal plane displayed on the left). In the vertical plane, the tilt relative to the fixation axis was small (approximately 1°, down) but approximately 3° (down) relative to the pupil axis (Fig. 5 , right). 

Discussion

The proposed procedure rests on the idea by Tabanero et al. ² that lens tilt can be measured by recording the fixation angle for which P3 and P4 are superimposed. With this procedure, only the direction of gaze has to be known; extensive calibrations to obtain a sequence of coefficients describing the linear changes of the Purkinje images with different eye positions can be omitted. As a potential improvement over the procedures introduced by Tabanero et al., ² a gaze tracker was incorporated in the current version that makes it unnecessary to present targets under known visual angles. 

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. 

Comparisons with Other Studies

Few studies provided data from both eyes of healthy subjects with phakic eyes. Rosales and Marcos ¹ state that “lens tilt tended to be mirror symmetric,” but they do not provide a detailed analysis. In addition, they do not describe interocular correlations in lens decentration and the striking lens decentration down from the pupil center that was found in the present study. Direct comparison between their data and the current data requires an interchange of x and y coordinates. If this were performed, their horizontal lens tilt would have ranged from −1.0° to +2.58° in the right eyes and from +0.66° to +1.99° in the left eyes. Their vertical lens tilt ranged from −1.13° to 2.80° in the right eyes to −1.96° to −2.87° in the left eyes. Although these values were within the range we observed, theirs averaged closer to zero and their interindividual variability was lower. More recently, ¹¹ they state that mirror symmetry was observed for tilt in the horizontal plane (as in this study) and decentration in the horizontal direction (not observed in this study; see Fig. 3E ). They also state that tilt was less than 2.6° in their study and that decentration was below 0.4 mm. Future measurements, perhaps in the same subjects with both techniques, can clarify whether there may be population differences or technical differences. 

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. 

Appendix 1

To understand the signs of the measured variables, it is important to recognize that the pixel coordinates in the video frame are x = 0 and y = 0 for the upper left edge of the computer screen and x = 752 and y = 536 for the lower right (European analog PAL video format). Another important fact is that the video frame display on the computer screen is not a mirror image of the subject but, rather, reverses left and right, reversing also the signs of both lens tilt and decentration measurements in the horizontal plane. 

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) × Hirschberg ratio/image magnification; kappa vertical = (y_P1 − y_pupil_center) × 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. 

 Submitted for publication August 8, 2007; revised November 22, 2007; accepted March 13, 2008.

 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.

 

Figure 1.

 

(A) Schematic drawing of the setup and technical details of the camera, lens, fixation LED, and IR LED for the generation of the Purkinje images. (B, left) Photograph of the setup. Right: appearance of the three Purkinje images in the pupil (P1 refers to the first, corneal Purkinje image, P4 and P3 to the Purkinje images from the back surface and the front surface of the lens, respectively). (C) Screen output of the software after data collection. Top right: regression of the distances from P3 to P4 versus gaze position is displayed separately for the x and y directions. Bottom right: ring carrying the LEDs in front of the camera lens is schematically shown. The position of the fixation point of the subject is continuously plotted during the measurements. It can be seen that the subject fixated the green fixation LED (see above left) and then looked at the numbers 2 and 14.

Figure 2.

 

Sign conventions used by the software. The measured subject’s eye is seen from above. The same sign conventions applied for left and right eyes, although right refers to temporal in the right eye and nasal in the left eye. PNP, posterior nodal point (approximate position). Total time for the completion of a measurement in one eye was approximately 1 minute.

Figure 3.

 

Comparisons of the kappas (A, B), lens tilt (C, D), and lens decentration (E, F) in both eyes of each of the measured subjects. In the horizontal plane (A, C, E), kappa and lens tilt displayed significant mirror symmetry, but lens decentration was not correlated. In the vertical plane, kappa, lens tilt, and lens decentration were positively correlated. Error bar denotes standard deviations from three repeated measurements in the same eyes. Orthogonal regression analysis was applied to study the correlations between both eyes. Slopes, intercepts, and correlation coefficients are shown in the figures.

Figure 4.

 

Lens tilts (A) and decentrations (B) in the horizontal and vertical directions in both eyes of the 11 subjects. Error bars denote standard deviations from three repeated measurements.

Figure 5.

 

Summary of kappa, lens tilts, and decentrations in the 11 subjects of the present study. The observed trends are emphasized by black arrows. The black spots on the lenses denote their geometrical center, approximating them as ellipses. Fixation axes and pupil axes are also denoted.

Figure 6.

 

Measurements in the 11 subjects with two different copies of the setup after 6 weeks. Note that the measurement results were reproducible.

The author thanks Juan Tabanero, Pablo Artal, Patricia Rosales, and Susana Marcos for demonstrating their devices before this study was initiated, and Hakan Kaymak and Ulrich Mester for stimulating this study.