fax: (812) 855-6616
email: thibos
indiana.edu
Address for correspondence:
Larry N. Thibos
School of Optometry
Indiana University
Bloomington, IN 47405
voice: (812) 855-9842 or 855-4475
fax: (812) 855-6616
email: thibosindiana.edu
key words: chromatic aberration, vision, pupil
Subjective transverse chromatic aberration (sTCA) manifest at the
fovea was determined for a population of 85 young adults (19 - 38
yr.) using a two-dimensional, two-color, vernier-alignment technique.
The statistical distribution of sTCA was well fit by a bivariate
Gaussian function with mean values which were not significantly
different from zero in either the horizontal or vertical directions.
We conclude from this result that a hypothetical, "average eye"
representing the population mean of human eyes with medium sized
pupils is free of foveal sTCA. However, the absolute magnitude of
sTCA for any given individual was often significantly greater than
zero, and ranged from 0.05 - 2.67 arcmin for the red and blue light
of a computer monitor (mean wavelengths 605nm and 497nm). The
statistical distribution of the absolute magnitude of sTCA was well
described by a Rayleigh probability distribution with mean = 0.8
arcmin. A simple device useful for population screening in a clinical
setting was also tested, and gave concordant results. Assuming that
sTCA at the fovea is due to decentering of the pupil with respect to
the visual axis, we infer from these results that the pupil is, on
average, well centered in human eyes. The average magnitude of pupil
decentration in individual eyes is less than 0.5 mm, which
corresponds to 
= 3 degrees for the angle between the achromatic and visual axes of
the eye.
The importance of the eye's chromatic aberration and its influence on visual performance is well documented. 1-7 A systematic description of the impact of ocular chromatic aberration on the retinal image includes three major optical effects: chromatic difference of focus, chromatic difference of position, and chromatic difference of magnification. The first of these is commonly called longitudinal chromatic aberration (LCA) whereas the other two are aspects of transverse chromatic aberration (TCA). Although early studies emphasized the defocusing effects of LCA on visual performance, more recent experiments have demonstrated the equally important effects of TCA on contrast sensitivity, localization of visual direction, color vision, and stereopsis. 5,7-13
In a simple optical model of the eye, the magnitude of TCA varies linearly across the retina and is zero at a single retinal locus. 4 The location of this singular, achromatic point depends upon the position of the pupil along each of the three cardinal directions (anterior-posterior, superior-inferior, nasal-temporal). Since the retinal achromatic point is unlikely to fall precisely in the center of the fovea, we must presume that central vision is subjected to some degree of TCA. In order to assess the functional impact of this TCA on vision, we need to quantify the magnitude of TCA present at the fovea. For this purpose it is convenient to define a secondary reference axis called the foveal achromatic axis, so named because it defines the path of light rays which pass through the eye to the fovea without suffering the dispersive effects of chromatic aberration. Since this path is also a reasonable operational definition of the visual axis, 4 we adopt the latter term in this report. Given these definitions, it follows that foveal TCA arises whenever the visual target is located off the visual axis or whenever the observer views a target through an artificial or natural pupil which is not well centered on the visual axis. 3,4,12,14 We take advantage of these facts in the present study to draw inferences about pupil centration from measurements of foveal TCA.
Aside from its importance as an optical element in the visual system, the pupil also has clinical significance as a diagnostic tool in the clinical practice of optometry and ophthalmology. 15 Although much is known about how pupil diameter varies with retinal illumination, accommodation, psychological state, and pharmaceutical agents, comparatively little has been published about the pupil's position within the eye, or how well it is centered with respect to the other optical elements of the eye. 2 Cross 16 states that large misalignments of the pupil center (corectopia) are very rare and small amounts of pupil misalignment are of no clinical significance. Nevertheless, moderate decentrations will influence results of the Hirschberg test and may affect the value of the angle lambda found from this test. Detecting errors of this kind is important for defining the optical zone of the cornea in refractive surgery.
Given the significance of foveal TCA for spatial and color vision, and the clinical significance of pupil centration, there is a need for information regarding the distribution of TCA and pupil location in the general population. Two recent studies of foveal TCA in a small sample reported values of just a few minutes of arc or less. 4,12,17 The implication of these results is that the natural pupil is remarkably well centered on the visual axis, suggesting the possibility of an adaptive optimization of pupil position to maximize image quality for foveal vision. Before pursuing the numerous implications of this intriguing finding, it is important to verify the result on a larger population. The purpose of the present study, therefore, was to determine if similar results obtain for a relatively large group of healthy, young adults. A secondary purpose was to test the utility of a simple, inexpensive screening device which may be suitable for extensive population studies or routine clinical use.
Although TCA is an optical phenomenon which is independent of the neural portion of the visual system, the manifest TCA measured by psychophysical techniques varies with the diameter of a displaced artificial pupil used to induce the aberration. 11,18-20 This effect is not expected from a simple, geometrical optics analysis of the eye's optical system. Instead, it has been attributed to the wave-guide properties of the cone photoreceptors which are responsible for the Stiles-Crawford effect (SCE). 18,20 This intrusion of the SCE complicates the interpretation of psychophysical measurements of chromatic aberration and requires that we distinguish clearly between the optical aberration (TCA) and its subjective manifestation (sTCA). Given this distinction it becomes possible to discuss the potential impact of the SCE on the interpretation of our results.
Foveal sTCA was measured with a two-dimensional version of the two-color vernier-alignment technique described in detail previously. 4 The principle of the method is that horizontal displacement of the pupil from the visual axis will induce ocular TCA at the fovea, causing a collinear pair of vertical lines, one red and one blue, to appear misaligned. Conversely, vertical displacement of the pupil will cause horizontally-oriented, collinear lines to appear misaligned. By arranging to have vertically-oriented lines and horizontally-oriented lines in the same target, it is possible to measure TCA simultaneously in the two directions. A detailed discussion of the theoretical framework which underlies the interpretation of data collected by this method is described elsewhere 4.
Subjects were recruited from the student body, faculty, and staff of the School of Optometry at Indiana University. From 164 volunteers, 99 subjects were selected based on refractive error criteria described below. Of these, 14 individuals were excluded on the basis of amblyopia, anisometropia, or age. The remaining 85 individuals constituted a homogeneous population of young adults (age range 19-36 years, average = 24.8 yr., s.d. = 3.73 yr.) with normal vision.
Inclusion criteria based on refractive error were as follows. (1) Subjects must have less than 0.5 diopters of astigmatism. (Rationale: astigmatism remained uncorrected during the experiment to avoid introducing unknown amounts of chromatic aberration with spectacle lenses, and it was feared that too much astigmatism might cause excessive errors in the two-color vernier-alignment procedure used to measure TCA.) (2) Hyperopes must have less than 2 diopters of hyperopia. (Rationale: hyperopes must accommodate even for a distant stimulus, and excessive accommodation may have unknown effects on pupil centration and lenticular TCA. ) (3) Myopes must have no more than 4 diopters of myopia. (Rationale: myopic refractive error was corrected with the appropriate negative-power, achromatic, cemented doublet lens selected from a set (power = -1, -2, -3 or -4 diopters; diameter = 7 cm) especially designed and fabricated for this experiment. 21)
We verified by the following procedure that the achromatic doublets used to correct the refractive error of myopic subjects did not introduce significant amounts of additional chromatic aberration during the experiment. An experienced observer familiar with the measurement of TCA viewed the two-color vernier alignment target (described below) through the edge of the lens. This worst-case scenario induced no detectable offset of the targets, which implies that any chromatic aberration induced by the lens was negligible. To be quite certain that the lenses would induce less than negligible amounts of TCA during the experiment, subjects viewed through a 1 cm diameter aperture centered on the lens.
Two versions of the two-color vernier-alignment target (Fig. 1) were used. The first was a simple, inexpensive, static device constructed from Wratten (Kodak, Inc.) gelatin filters (red: #92, mean wavelength = 651 nm; blue: #47B, mean wavelength = 454 nm) and transilluminated with white light from a tungsten lamp. In this context we define mean wavelength of a polychromatic light source as the first moment of its luminance spectrum. As illustrated in Fig. 1A, the target consists of a red cross accurately aligned with abutting, blue extensions. Using the process described in detail by Bradley (see Fig. 3 from Bradley 13 ), we calculated the chromatic difference of refractive error (i.e. the longitudinal chromatic aberration of the eye) for these red and blue components of the target for the Chromatic Eye model of ocular chromatic aberration 22 and the result was 1.20 D. (To make this calculation we used published specifications for the filters and assumed the tungsten lamp had color temperature 2800 K.) The advantages of this target are low cost, large chromatic difference of refractive error, narrow spectral bandwidth of the two colors, and speed of testing. As such it makes a useful screening test which may prove suitable for routine clinical practice in the future. The disadvantage is that it allows measurement of only the sign of sTCA, not its magnitude. In the balance of this paper we will refer to the device as a "TCA detector".
Figure 1. Visual stimuli used to measure transverse chromatic aberration were based on Wratten filters (A,B) or a computer monitor (C,D). Top row is the target, bottom row is the subjective appearance of the target. A. Physical configuration of strips of red (hatched) or blue (stippled) filter in an opaque mask. B. Subjective appearance of target A when viewed through an artificial pupil displaced up and to the right with respect to the visual axis. C. Physical location of black bars on colored backgrounds required so they will appear to be aligned. D. Subjective appearance of target in C when viewed through an aperture displaced up and to the right. Drawings are not to scale.
If an individual reports a misalignment of the TCA detector in a free-viewing situation, we attribute that result to foveal TCA caused by decentration of the natural pupil with respect to the visual axis. Accordingly, we adopted a sign convention for this test based on the appearance of the TCA detector when viewed through a displaced artificial pupil. When the artificial pupil is displaced upward and to the right, optical TCA is induced at the fovea which causes the vernier target to appear misaligned in the directions indicated by Fig. 1B. To preserve bilateral symmetry, we used a mirror-symmetric sign convention which assigns positive values to the horizontal TCA induced by temporal displacement of an artificial pupil or to the vertical TCA induced by superior displacement of the pupil. Accordingly, if a subject reported misalignment of the TCA detector in the direction shown in Fig. 1B when viewing with the right eye, then we recorded it as positive TCA in the horizontal and vertical directions and we interpreted the result as a displacement of the natural pupil in the superior and temporal directions. However, if the same report was given when viewing with the left eye, then we recorded it as negative TCA in the horizontal direction, positive in the vertical direction, and we interpreted the result as a displacement of the natural pupil in the superior and nasal directions.
In order to measure the magnitude of sTCA as well its direction, another version of the two-color vernier target was presented on a standard color monitor under computer control (Macintosh II, by Apple Inc.). The stimulus was a red square surrounded by a blue background, with black reference lines placed at fixed locations on the blue field and a crossed pair of moveable black lines inside the red square (Fig. 1C). The vertical position of the adjustable horizontal line, and the horizontal position of the adjustable vertical line, were controlled by the subject using the computer's mouse. When viewed through an artificial pupil displaced upward and to the right, TCA will cause the black vernier lines to appear misaligned even though they are physically aligned. To compensate for the TCA induced by the aperture, the subject would have to shift the black lines contained within the red square by an equal and opposite amount until both the vertical and the horizontal lines of the cross appeared to be aligned with the corresponding reference lines within the blue square, as illustrated in Fig. 1D. The amount of physical misalignment introduced into the target by the subject to achieve subjective alignment is therefore a measure of foveal sTCA in object space. The sign convention is the same as described above for the static TCA detector. For example, the target illustrated in Fig. 1C corresponds to positive sTCA in the horizontal and vertical directions when viewing with the right eye, but corresponds to negative sTCA in the horizontal direction and positive sTCA in the vertical direction when viewing with the left eye.
The advantage of this computerized apparatus is that repeated, quantitative measures of sTCA could be obtained by asking the subject to adjust the target until it appeared aligned both vertically and horizontally. A disadvantage is the relatively small spectral separation between the red (mean wavelength = 605 nm) and the blue (mean wavelength = 497 nm) targets. Colored regions of the target were generated by exciting a single phosphor and the spectral energy distribution of these phosphors was calibrated with a spectral photometer (SpectraScan PR714 by Photo Research, Inc.). A detailed description of this calibration procedure for the monitor used in the present experiment is provided elsewhere. 23 The chromatic difference of refractive error determined for this target was 0.65 D for the Chromatic Eye model.
For the Chromatic Eye model the rate of change of TCA with pupil
displacement is equal to the chromatic difference of refractive
error, 
R.
4,22
 |
|
Accordingly, 1.2 D of chromatic difference of refractive error calculated for the TCA detector should cause TCA to change at the rate of 1.2 radians/meter, or 4.1 arcmin/mm of pupil displacement. For the computer monitor stimulus, a 0.65 D chromatic difference in refractive error should cause TCA to change at the rate of 0.65 radians/meter, or 2.2 arcmin per mm of pupil displacement. This expected value overestimated by about 20% the empirical results of a pilot study involving 6 observers in which subjects viewed the computer monitor through a 1mm pinhole. The mean rate of change of TCA induced by displacing a 1 mm pinhole from the visual axis was 1.8 arcmin/mm (s.d. = 0.2), which is equivalent to 0.53 diopters (s.d. = 0.06) of chromatic refractive error. We suspect this minor discrepancy was due to individual differences since the same procedure using the same apparatus performed previously on a different group of 5 individuals yielded a mean rate of change equal to 2.19 arcmin/mm. 23
Since previous experiments have indicated that foveal sTCA of the natural eye is commonly less than 1 arcmin 4,12 , the present experiments were designed to measure sTCA with significantly less uncertainty than this value. The position of the adjustable vernier lines on the computer monitor could be controlled to within one pixel (1/72 inch) and so a viewing distance of 5 m provided measurement resolution of 15 arcsec. This value was slightly less than the standard deviation of vernier acuity settings for the two-color alignment task in a pilot experiment conducted to estimate experimental variability (mean standard deviation of 20 settings for each of 6 subjects, including naive and experienced observers, was 20 arcsec). Assuming that the settings of the study population of untrained observers would also have a standard deviation of about 20 arcsec, N= 25 repeated measures of vernier alignment for each subject would be expected to yield a 95% confidence interval of ± 8 arcsec about the mean, thus satisfying the requirement for high precision. When expressed in terms of pupil decentration, this confidence interval corresponds to a measurement uncertainty of ± 0.06 mm of displacement of the natural pupil from the visual axis.
The TCA detector is more sensitive (because of the larger wavelength separation) than the computerized apparatus, but does not have the statistical advantage of repeated measures since observers are unable to make multiple, independent judgments about the alignment of a static display. These two factors roughly balance, thus yielding about the same level of sensitivity to pupil displacement for the TCA detector (threshold vernier acuity of 20 arcsec corresponds to 0.08 mm pupil displacement, by Eq. 1) as for the computer apparatus.
Pupil diameter during experiments was measured with a video camera mounted near the subject's head which focused on the subject's eye through a beam splitter. In order to minimize chromatic dispersion caused by the thickness of the beam splitter, the splitter was oriented almost perpendicular to the subject's line of sight (Fig. 2). We verified that the beam splitter induced no detectable offset of the vernier targets, which implies that any chromatic aberration induced by the monitoring system was negligible. Frequent checks were made to ensure that the subject's eye was properly aligned with the apparatus and that the lids were not vignetting the pupil. Quantitative measures of pupil diameter were taken off the calibrated face of the video monitor. This calibration was determined separately for each achromatic doublet lens used in the experiment to account for changes in optical magnification. An average of 5 pupil diameter readings was taken during each session and the average pupil diameter across all subjects was 4.7 mm. Fluctuations of more than 1 mm were rare, and TCA data were discarded if pupil variations of more than 2 mm were detected.
Figure 2. Experimental apparatus. Pupil and lids were monitored with a video camera and semi-transparent mirror S. Myopic subjects required refractive correction with achromatic lens F, masked by aperture A. Viewing distance for stimulus was 5 m.
Commercially available software (Systat) was used for statistical analysis 24,25 of the four variables measured for each subject (horizontal TCA in the left eye, vertical TCA in the left eye, horizontal TCA in the right eye, vertical TCA in the right eye). All hypothesis testing was conducted at the 5% significance level.
Tables 1A and 1B show the incidence of sTCA in the population of left and right eyes, respectively, as determined by the TCA detector constructed from colored filters. Each value in these tables represents the number of individuals reporting offset of the two-color vernier target in the direction specified. For example, when testing the left eyes of our 85 subjects, 3 individuals reported target displacement consistent with superior nasal displacement of the pupil. Inspection of the data in Tables 1A, 1B revealed no systematic differences between the left and right eyes and so the data were combined for further analysis. For the pooled data set, 70% (120/170) of eyes had no detectable sTCA in the vertical direction, 62% (106/170) had no detectable sTCA in the horizontal direction, and 48% of eyes (81/170) had no detectable sTCA in either direction. For the remaining 52% of eyes with detectable levels of foveal sTCA in at least one direction, positive and negative values were equally likely in both the horizontal and vertical directions. We infer from these results that for half of the eyes tested the natural pupil was so well centered on the visual axis that sTCA was below psychophysical threshold. For the other half, the pupil appeared to be randomly distributed about the visual axis in both the vertical and horizontal directions.
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The scatterplot in Fig. 3 shows the two-dimensional distribution of foveal sTCA measured quantitatively with the computerized apparatus (circles = left eye, crosses = right eye). Abscissa values are sTCA measured in the horizontal direction (i.e. horizontal misalignment of vertically oriented bars) and the ordinate values are sTCA measured in the vertical direction (see Methods for sign convention). Each data point in the scatterplot represents the mean of 25 settings for a particular individual. Although we designed the experiment based on pilot data so that the 95% confidence interval for each mean value would be about ± 8 arcsec (see Methods), the variance of individual data sets turned out to be 50% larger than expected on average (30 arcsec) which raised the average confidence interval to about ± 12 arcsec. Given this result, the reader may visualize the 95% confidence region for a typical data point in Fig. 3 as a circular region concentric with the plotted symbol and about 3 times the diameter of the symbol. Confidence regions for only 10 of the 170 eyes contained the origin in Fig. 3, which implies that most of the eyes tested had a statistically significant degree of sTCA in one or both directions.
Figure 3. Scatter plot and frequency histograms of the distribution of foveal sTCA measured in left and right eyes of 85 normal, healthy individuals. Covariance matrix for the pooled dataset is inset at upper left (correlation coeff. = 0.21). Means and standard deviations of the vertical and horizontal components of sTCA are given separately for the two eyes in table at bottom. Ellipse is centered on the mean and encompasses 50% of the probability of a bivariate Gaussian function fit to the data.
Frequency histograms of the horizontal components of sTCA for the two eyes are shown at the bottom of Fig. 3 and the frequency histograms for the vertical components of sTCA are shown at the right of the figure. Only one of these four histograms (horizontal sTCA, right eye) had a mean value (0.31 arcmin) which was significantly different from zero (t-test, see Methods). Pairwise comparisons of the four experimental variables measured for each subject (horizontal and vertical sTCA for each of two eyes) revealed only one statistically significant correlation (F-test): the vertical component of sTCA was positively correlated (r=0.6) in the two eyes.
Since the differences between sTCA for the left and right eyes were not large, the data were pooled for further analysis to be described below. Before performing this analysis, however, we checked whether our choice of sign convention for the horizontal component of sTCA mattered. This was done by computing the variance of the pooled data sets using a left-right convention and again using a nasal-temporal sign convention. No significant difference in these two variances was found (F-test) and therefore we chose to retain the body-symmetric sign convention described in Methods.
The average abscissa value for the combined population of 170 eyes was +0.073 arcmin (s.d. = 0.61 arcmin) and the average ordinate value was -0.009 arcmin (s.d. = 0.75 arcmin). Neither of these mean values was significantly different from zero, which leads us to conclude that a hypothetical "average eye" representing the central tendency of the population of young adult eyes with medium sized pupils is free of foveal TCA. The correlation (r=0.12) between vertical and horizontal components of sTCA lacked statistical significance but the small difference between variances of vertical and horizontal components of sTCA was marginally significant. The covariance matrix for the pooled dataset is provided as an inset table in the scattergram of Fig. 3.
The frequency histograms for vertical and horizontal components of sTCA were well fit by one-dimensional Gaussian probability functions. The Pearson chi-squared test of normality performed separately for these two components of sTCA indicated that in both cases there was no basis for rejecting the hypothesis that the population was normally distributed. This conclusion was confirmed by a second test of normality in which the cumulative histograms (after pooling right and left eyes) were plotted on probability axes. The results were well fit by linear functions (least-squares regression, r > 0.98). Since both marginal distributions were found to be Gaussian, we infer that the two-dimensional frequency distribution of sTCA at the fovea may be described as a two-dimensional Gaussian. The ellipse shown in Fig. 3 is centered on the mean and encompasses 50% of the probability of the bivariate Gaussian function fit to the data. Thus it is equally likely that the data point for an individual would fall inside or outside this curve.
In some applications (e.g. computing retinal image quality) the sign of sTCA is immaterial, so there is reason to examine the statistical distribution of its magnitude, |sTCA|. This positive quantity is given by the radial distance from the origin to a given data point in Fig. 3 (i.e. the square root of the sum of squared components of sTCA in the horizontal and vertical directions). Fig. 4 shows the frequency distribution of |sTCA| values for our population of 170 eyes. The mean of this distribution is 0.83 arcmin (s.d. = 0.53 arcmin) which indicates that, on average, individual eyes in the test population suffered from less than 1 arcmin of foveal sTCA.
Figure 4. Distribution of the magnitude of subjective TCA as derived from the two-dimensional scatterplot in Fig. 3. The experimental distribution is well described by a theoretical Rayleigh distribution of the same mean and standard deviation as the empirical data.
In theory, if the two-dimensional distribution of sTCA in
Fig. 3 is Gaussian with zero mean and
standard deviation S in both the horizontal and vertical
directions, then the distribution of |sTCA| in Fig.
4 should have a Rayleigh distribution with mean =
S(
/2)1/2.
Although the measured variances of sTCA for horizontal and vertical
directions were statistically different, this difference was slight
and did not disrupt the excellent fit of the theoretical Rayleigh
distribution to the experimental histogram as shown in Fig.
4. Nevertheless, a rigorous test of this hypothesis required
that the individual values of sTCA in the horizontal and vertical
directions be normalized by the corresponding standard deviation
prior to performing a Pearson chi-squared test of goodness-of-fit
(p=0.05). The results of this test confirmed that the distribution of
normalized |sTCA| is well fit by the Rayleigh distribution, as
expected for a bivariate Gaussian variable.
Since the TCA detector and the computerized test provide independent estimates of the sign of sTCA for each individual in the study, it was of interest to see if the two test results were in agreement. To put the data from the two experiments on a common basis for comparison, positive values of sTCA were encoded as +1, negative values as -1, and the absence of sTCA was encoded as 0. This encoding scheme was applied independently to the horizontal and the vertical components of sTCA. In order to decide which values of sTCA measured with the computerized stimulus were functionally insignificant and therefore to be encoded as 0, we appealed to the statistical analysis described above in connection with Fig. 3 which indicated that a 95% confidence interval for individual means was typically about 0 ± 0.2 arcmin. Accordingly, since any measured value of sTCA in the range (-0.2, +0.2) arcmin is not significantly different from zero, we assigned it the zero value.
Comparison of the results of the two experimental methods is given separately for the horizontal and vertical components of sTCA in Tables 2A, and 2B, respectively. Each table cell contains the number of occurrences of the particular combination of results specified by the row and column headings of the table. For example, for 21 individuals both methods agreed that horizontal sTCA was negative, but for 2 individuals the TCA detector indicated positive TCA when the computer monitor test indicated negative TCA. A similar pattern of results emerged for the horizontal and vertical components of sTCA and so Tables 2A and 2B were pooled to form Table 2C. The main diagonal of the matrix of numbers in Table 2C represents the number of times the results of the two tests agreed, which in this study accounts for 41% of the comparisons. Opposite results on the two tests occurred only 6% of the time and the remaining 53% of comparisons were mainly cases where the TCA detector revealed no misalignment but the computer monitor test revealed a statistically significant value of sTCA. Evidently the computerized test is the more sensitive of the two since there were far fewer eyes for which sTCA was judged to be zero. This superior sensitivity appears to be due to the use of repeated measures. In principle, if the number of measurements N is large enough, the 95% confidence interval will be vanishingly small and therefore almost no eyes will be classified as having zero sTCA. This line of reasoning suggests that there exists some smaller value of N for which the two tests would be equally sensitive. This was borne out by additional re-coding analysis which indicated that for N=3 measurements (i.e. confidence interval = ± 0.6 arcmin), the frequency distribution for sign(sTCA) is nearly the same for the two tests. Thus the advantage of a large wavelength difference in the TCA detector is equivalent to the statistical advantage of just a few repeated measures.
A: Horizontal sTCA
B: Vertical sTCA
C: Combined V&H
To assess the repeatability of foveal TCA measurements, a random subgroup of 22 subjects was selected for a second measurement of the horizontal component of sTCA using the computerized test. The two sessions were separated by several weeks and the comparison of results is drawn in Fig. 5. Correlation coefficients were greater than 0.7 for both the left and right eyes and the least-squares regression lines had slopes that were not significantly different from one and intercepts not significantly different from zero. Thus we conclude that repeatability of sTCA measurements with the computer monitor stimulus is reasonably good even for untrained subjects.
Figure 5. Repeatability of measurements of the horizontal component of subjective TCA. From participants in the main experiment (trial #1), 22 individuals were randomly selected for a second set of measurements taken several weeks later (trial #2). The diagonal reference line has unity slope and passes through the origin. Inset shows frequency distribution of the difference between first and second measurements of sTCA (mean = -0.10, st.dev. = 0.49).
On the other hand, the discrepancy between test and re-test results was sometimes more than could be accounted for by measurement variability. The inset histogram of Fig. 5 shows the frequency distribution of differences in sTCA measured at the two sessions. This difference was greater than the expected 95% confidence interval (± 0.2 arcmin) for mean sTCA in more than half (23/42) of the subjects. From this result we conclude that for the majority of individuals examined, some of the test/re-test discrepancy was due to small physiological changes in the optical configuration of their eyes which occurred between experimental sessions.
Previous estimates of foveal sTCA obtained from a small number of highly trained observers have indicated that values greater than 1 arcmin, even for widely spaced wavelengths, are uncommon. 4,12 The present experiment confirms that result for a relatively large population of 170 eyes in 85 young adult observers. The low variance and high repeatability of results obtained from untrained individuals indicates that accurate assessment of foveal sTCA in a clinical setting is a reasonable expectation. To this end, a simple screening device designed to detect the presence of significant amounts of foveal sTCA was built from inexpensive components, tested experimentally, and shown to be nearly as sensitive as more complex laboratory equipment.
Previous experiments have shown that controlled amounts of TCA may
be induced at the fovea by displacing an artificial pupil from the
visual axis. 4,12
This result is well described by a simple, reduced schematic-eye
model which, in the paraxial region, predicts the linear relation
embodied in Eq. 1 above. 4,22
Thus, the simplest explanation for the existence of residual amounts
of foveal TCA under free viewing conditions is that the eye's natural
pupil is slightly displaced from the visual axis. Assuming this is
the case, we may infer the amount of pupil decentration simply by
dividing the measured amounts of sTCA by R,
the chromatic difference of focus, which has a value of 2.2 arcmin/mm
for the computer monitor used in the present experiments (see
Methods). Thus the bivariate Gaussian distribution of sTCA shown
in Fig. 3 may be interpreted as a
distribution of pupil locations simply by rescaling the coordinate
axes as shown in Fig. 6.
Figure 6. Distribution of pupil displacement from the visual axis (bottom and left axes) and distribution of angle psi (angle between visual axis and achromatic axis of the eye, top and right axes) inferred from data of Fig. 3. Ellipse encompasses half of the data points.
When considering the population as a whole, positive and negative values of pupil decentration tend to cancel and thus the statistical distribution of pupil centers has zero mean. Thus for the purposes of defining a hypothetical "average eye" which represents the population mean, the present results justify assuming that the pupil is well centered on the visual axis. This remarkable result, that central vision tends to be spared the deleterious effects of TCA which exist everywhere else in the visual field, was discovered first in a small group of graduate students and vision scientists, 4,12 and now is seen to hold also in a larger population consisting mainly of young optometry students. Perhaps this fortunate optical configuration in human eyes is the result of an active process or evolutionary adaptation which has maximized retinal image quality where it would do the most good, namely, in that part of the visual field where neural resolving power is maximum.
On the other hand, for any particular eye the pupil may be displaced from the visual axis by a significant amount. Regardless of the direction of displacement, the resulting TCA and other transverse, monochromatic aberrations will attenuate retinal image contrast and jeopardize spatial vision. 10,26 Therefore, to appreciate the functional impact of pupil decentration in the population, it is more instructive to inspect the distribution of absolute values shown in Fig. 4. When the abscissa of that figure is rescaled by the conversion factor 2.2 arcmin/mm, the mean pupil displacement is found to be 0.37 mm with a standard deviation of 0.24 mm. Although these values are small in comparison to other ocular dimensions, the amount of TCA induced may nevertheless be functionally significant. A mean foveal TCA of 0.82 arcmin is several times larger than vernier acuity, for example, and represents a full 180 degrees of phase shift between the red and blue components of a 37 cyc/deg polychromatic grating. Perhaps the largest visual effect will be on binocular depth perception. Simonet and Campbell have emphasized that binocular vision is so sensitive to small amounts of foveal TCA that even sub-threshold levels of TCA can lead to measurable stereoscopic effects. 11 We pursue this point further in the next section.
In a binocular viewing situation, equidistant objects of different color often appear to lie at different distances, with red objects generally appearing closer than blue when viewed against a black background. 27 This phenomenon, called chromostereopsis, can be experimentally induced by displacing artificial pupils in front of each eye in a temporal direction (red objects will appear closer) or in the nasal direction (blue objects will appear closer). A simple optical model accurately accounts for both the magnitude and direction of this effect by assuming that the binocular disparity which underlies the sensation of depth is due to the algebraic sum of the monocular sTCA induced in the two eyes by the displaced pupils. 28 Assuming this model holds also for sTCA induced by displacement of the natural pupil, we calculated the binocular disparity expected for each subject by summing the horizontal components of sTCA in the two eyes. A frequency histogram of the results is shown in Fig. 7. The mean, mode, and median of this distribution are all positive, which is consistent with previous reports that most people see red objects in front of blue. 29,30 However, there is considerable variance in the distribution of Fig. 7 and a significant number of subjects had negative binocular disparity, which is consistent with reports that some subjects report seeing blue in front, especially at low luminance when the pupil is larger. 11,30,31 The possibility that factors other than TCA may also influence chromostereopsis under some stimulus conditions is an issue of current interest. 23,32
Figure 7. Frequency distribution of computed binocular disparity for red and blue targets (T) displayed on computer monitor. Disparity was calculated as the sum of monocular sTCA for the left (L) and right (R) eyes of each individual. Positive disparity corresponds to bi-temporal displacement of the pupils, which causes apparent location of red target (open circle) to be closer than blue target (closed circle) as illustrated. Solid lines indicate refracted light rays, dashed lines indicate projection of retinal images through nodal point N into object space. Negative disparity correspond to bi-nasal displacement of the pupils, which causes blue to appear closer than red.
Previously we defined
as the angle between the eye's achromatic axis (the primary reference
axis in the theory of ocular chromatic aberration, defined as the
line joining the nodal point and pupil center) and the visual axis
(i.e. foveal achromatic axis, the line joining nodal point and
fovea). 4 For the chromatic eye model,
the angle
is related to pupil displacement from the visual axis by the
equation
 |
|
where NP is the distance between first nodal point and the plane
of the entrance pupil. 4 Combining Eqs.
(1) and (2) and
employing the small angle approximation sin()
=
gives
 |
|
Therefore, assuming the customary value 10
NP = 3.98mm and taking R
= 2.2 arcmin/mm, we can interpret Fig. 3 as
a frequency distribution of
in degrees of visual angle if we rescale the axes by multiplying by
the factor 6.5 as shown in Fig. 6. Although
our previous publications considered only the horizontal component of
,
it is now clear that the vertical component is of the same order of
magnitude.
The foregoing inference of pupil position from measurements of TCA
does not distinguish between the optical aberration (TCA) and its
subjective manifestation (sTCA). To account for the empirical
observation that sTCA can be considerably less that TCA, Ye et al.
suggested that the Stiles-Crawford effect (SCE) behaves like a neural
weighting function impressed upon the blurred retinal image.
20 The effect of this weighting
function is to reduce the apparent difference of position for blurred
retinal images containing light of different wavelengths. However,
the influence of the SCE in this regard depends critically upon its
symmetry in the pupil plane. It is helpful at this point for the
reader to conceive of the SCE as an apodization of the pupil.
33 If that apodization function has
even symmetry about the center of the pupil, then the SCE will have
no influence on the location of blurred retinal images, and therefore
will have no effect on the magnitude of sTCA (see Fig. 1 of Ye et al.
20 ) Consequently, the simple scheme
described above for inferring pupil position and angle
from measurements of sTCA applies not only when the SCE is
disregarded, but also when the SCE apodization function is symmetric
about the pupil center.
There is abundant evidence in the literature to support the assumption that the SCE is usually symmetric and well centered on the pupil, possibly because of a phototropic effect within the photoreceptors. 34-41 Nevertheless, exceptions to this rule are not uncommon and so there remains some uncertainty about the validity of inferences we have drawn from sTCA results since we neither measured the SCE nor determined its centration with respect to the natural pupil in the present experiments.
We thank Mr. Kevin Haggerty for technical support, Arthur Bradley for advice on experimental design and critical review of the manuscript, and the Statistical Consulting Service at Indiana University for advice on statistical design and data analysis. David Still contributed to the development of the TCA detector, also known as a hyperachromatical alignicator. This research was supported by NIH grant R01 EY05l09 to L.N.T.
