A Statistical Model of the Aberration Structure of Normal, Well-Corrected Eyes

Address for correspondence:

Larry N. Thibos
School of Optometry
Indiana University

Bloomington, IN 47405
thibosindiana.edu
812-855-9842

Submitted to Ophthalmic and Physiological Optics, 18 April 2002

Revised 14 May 2002.8 figures, 0 tables

ABSTRACT

A statistical model of the wavefront aberration function of the normal, well-corrected eye was constructed based on normative data from 200 eyes which show that, apart from spherical aberration, the higher-order aberrations of the human eye tend to be randomly distributed about a mean value of zero. The vector of Zernike aberration coefficients describing the aberration function for any individual eye was modeled as a multivariate, Gaussian, random variable with known mean, variance, and covariance. The model was verified by analyzing the statistical properties of 1,000 virtual eyes generated by the model. Potential applications of the model include computer simulation of individual variation in aberration structure, retinal image quality, visual performance, benefit of novel designs of ophthalmic lenses, or outcome of refractive surgery.

INTRODUCTION

Traditionally, schematic eye models have had fixed properties which represent the optical properties of the average human eye (Thibos and Bradley, 1999). Perhaps the most flexible model eye yet devised is the Indiana reduced eye (Thibos and Bradley, 1999), which allows for variation in the axial and lateral location of the pupil, which in turn allows for variation in the various reference axes of the eye and the magnitude of off-axis astigmatism. The model also allows for variation in the focal length and refracting power, which permits the modeling of axial and refractive focusing errors. By adjusting the shape of the aspheric refracting surface this model accurately describes the mean spherical aberrations and oblique astigmatism of eyes, but not the irregular, higher-order aberrations. Because of its flexible configuration, the Indiana Eye can provide customized models that describe specific individuals. Nevertheless, like all other schematic eyes in the literature, once configured it has fixed optical properties. Consequently, such models fail to capture the rich variation between individuals or the changes that take place over the lifespan, during accommodation, following refractive surgery, or as a result of normal biological fluctuations.

A second limitation of traditional schematic eyes is that they provide only indirect links between model parameters and optical performance. Most schematic eyes described in the visual optics literature require substantial optical analysis using sophisticated ray-tracing computer software to establish the link between structure and function. Instead, our goal was to define the model in terms of a wavefront aberration function (WAF) rather than as a physical arrangement of optical elements. In this way a direct link is established between the model and the retinal image produced by the model through the traditional methods of Fourier optics (Goodman, 1968).

The need for a statistical approach to modeling the eye was articulated first by Walsh, Charman, and Howland (Walsh et al., 1984) who said,

“Human eyes are neither radially symmetrical nor uniform. Their higher-order aberrations can no more be described by a single schematic eye than can their sphero-cylindrical prescriptions. … There is a rich variety of higher-order aberrations of the human eye, with the eyes of no two persons being exactly alike. The variety is so great and exists in so many dimensions that no single set of Taylor coefficients can meaningfully be said to be typical. Our use of mean Taylor coefficients and average critical pupil sizes is simply an attempt to characterize this variety and should not be interpreted as an effort to construct a schematic eye of higher-order aberrations.”

Inspired by these observations, we aimed to overcome the limitations of traditional schematic eyes by deriving a statistical model of the WAF. This is a departure from previous approaches in two ways. First, we sought a description of the aberrations of an eye, rather than the physical system that produced those aberrations. In this way, we aimed to build a functional model rather than a physical model of the eye’s optical system. Second, we envisioned the model as a stochastic process, rather than a system with fixed parameters, so that change and variation are an inherent feature of the model. Like Walsh, Charman and Howland before us, we view the WAF rather like fingerprints: each is unique and the population as a whole can only be described statistically.

METHODS

The statistical behavior we modeled was observed in the Indiana Aberration Study, which is described in detail elsewhere (Thibos et al., 2002). That study employed a Shack-Hartmann wavefront sensor to measure WAFs along the primary line-of-sight of 200 normal, healthy eyes from 100 individuals for a 6mm pupil. Most subjects were optometry students in the age range 22-35 yr. Since we were interested primarily in the higher-order aberrations of the eye, we paralyzed accommodation and corrected the sphero-cylindrical refractive errors with trial lenses. Small residual refractive errors were inevitable because the lenses come in 0.25D steps, but otherwise we think of these eyes as being well-corrected.

A Zernike expansion represents each WAF as a weighted sum of Zernike polynomials,

where W(x,y) is the wavefront defined over the x,y coordinates of the pupil, c is an aberration coefficient, and Z(x,y) is a Zernike polynomial. Thus, a compact way of describing the aberration structure of an eye is to quote the vector of aberration coefficients a_n^f used in a Zernike expansion of the WAF. Three such vectors, corresponding to three independent measurements, were averaged for each eye.

Zernike coefficients are specified in this report using the standard nomenclature defined with reference to the standard coordinate system recommended by the Optical Society of America (Thibos et al., 2000). In this coordinate system, the x-axis is horizontal pointing to the right from the viewpoint of the experimenter when facing the patient, the y-axis is vertical pointing up, and the z-axis is axial pointing away from the eye. Previous studies have shown that the aberration coefficients of the right and left eyes are statistically correlated due to a systematic tendency towards bilateral symmetry (Porter et al., 2001; Thibos et al., 2002). Bilateral symmetry in the aberration structure of eyes would make W(x,y) for the left eye equal to W(-x,y) for the right eye. In this case, the Zernike coefficients for the two eyes would have opposite sign for all those modes having odd symmetry about the y-axis (e.g. mode Z₂^-2 ). To take this symmetry into account prior to pooling of the results from left and right eyes, aberration coefficients for the right eye were converted to an equivalent set of coefficients for the left eye by the linear transformation L=M*R, where M is a diagonal matrix with elements +1 (no sign change) or –1 (with sign change). For example, matrix M for Zernike vectors representing the first 4 orders (15 modes) would have the diagonal elements [+1, +1, -1, -1, +1, +1, +1, +1, -1, -1, -1, -1, +1, +1, +1]. This conversion is equivalent to adopting a sign convention in which the positive x-axis points temporally rather than to the right.

RESULTS

Population statistics

The statistical behavior we modeled is summarized in a pyramid of frequency histograms shown in Fig. 1. Each histogram displays the frequency of occurrence of aberration value for a specific Zernike coefficient a_n^f. Each row of the pyramid corresponds to a given radial order n and each column corresponds to a given meridional frequency f of the Zernike polynomial. These data show that frequency histograms for the first 15 Zernike modes are bell-shaped and symmetric about the mean values for all modes. This suggests that Zernike coefficients could reasonably be described as Gaussian random variables. To visualize such a model, we overlaid each histogram with a Gaussian probability density function that has the same mean and variance as the experimental distribution as shown in Fig. 1. In each case the Gaussian model appears to provide an adequate description of the empirical data. A formal test of this hypothesis using the chi-squared statistic (p=0.05 significance level) indicated that most (30 of 36 modes) Zernike coefficients in our sample were well fit by a Gaussian model. We therefore adopted a multivariate Gaussian model for our statistical model eye.

One major advantage of a multivariate Gaussian model is its simplicity. The statistical structure of such a model is fully defined by the means and variances of the various Zernike modes and by the covariance between all possible pairs of modes. A graphical display of the vector of mean values is shown by the symbols in Fig. 2. The error bars in this figure indicate ± 2 standard deviations from the mean. Thus, aberration coefficients for individual eyes have a 95% probability of falling inside the ranges depicted by the error bars.

Two features of the data in Figs. 1 and 2 are noteworthy. First, for most Zernike modes the aberration coefficients are symmetrically balanced around zero. This result suggests that the central tendency for human eyes is to be free of higher-order aberrations, but that any given individual is equally likely to have positive or negative aberrations due to random biological variability. The one clear exception among the higher-order modes is spherical aberration a₄⁰, which is systematically biased towards positive values. Second, the range of values decreases systematically with mode number. This means that the magnitude of aberration coefficients in any given individual tends to be smaller for higher order modes than for lower order modes. Analysis presented elsewhere shows that for our population of eyes, average wavefront variance falls exponentially with radial order (Thibos et al., 2002).

Pearson correlation coefficients computed for each pair of Zernike modes through the 7th order are shown in Fig. 3 in the form of a gray-scale image. Only a few examples are found of positive correlations (indicated by light squares). The largest positive correlation coefficients are between vertical prism (n = 1, f = -1) and vertical coma (n = 3, f = -1; R = 0.88) and between horizontal prism (n = 1, f = +1) and horizontal coma (n = 3, f = +1; R = 0.76). The only other large positive correlation was between defocus (n = 2, f = 0) and spherical aberration (n = 4, f = 0; R = 0.48). All of the other large correlations evident in Fig. 3 are negative. Most prominent among these occurred between modes of the same frequency (e.g. n = 3, f = -3 vs. n = 5, f = -3; R = -0.36 or for n = 4, f = -4 vs. n = 6, f = -4; R = -0.48) or between modes of the same order (e.g. n = 3, f = -3 vs. n = 3, f = -1; R = -0. 53 or for n = 4, f = +2, vs. n = 4, f = +4; R = -0.40). All of this information about correlation between modes is contained in the covariance matrix that is part of the multivariate Gaussian model. \

Validation of the model

Our statistical model eye consists of a computer program that embodies the statistical structure described above. This program generates an arbitrary number of virtual eyes, in the form of a vector of Zernike coefficients, drawn from a population with known statistics. To validate the model, we configured it to produce 1,000 virtual eyes and then examined the results statistically. For example, in Fig. 4 we show the frequency distributions of individual aberration coefficients produced by the model (solid curves). These clearly fit the experimental frequency histograms well. The means and standard deviations of each of these histograms, plus additional data from modes through the 7th order, are compared in Fig. 5. If the model accounts well for the human data, then each data point (corresponding to one mode) will fall somewhere along the dashed line. Clearly this expectation is fulfilled for both statistical parameters (R > 0.99).

Scatter plots are useful for visualizing the statistical correlation between various pairs of modes. For example, in Fig. 6A we show the positive correlation between the defocus coefficient a₂⁰ and the spherical aberration coefficient a₄⁰. The dashed curve in this figure is a 95% probability ellipse (i.e. individual data points have a 95% probability of falling inside the ellipse) for a bivariate Gaussian distribution with the same mean, variance and covariance as the experimental data. A similar analysis performed on the aberration coefficients of the virtual eyes produced by our computer model are shown in Fig. 6B. A visual comparison of these two graphs indicates that the correlation embodied in the probability ellipse is equally as effective in describing the human eye data as the model eye data. An example of negative correlation is shown in Fig. 7, this time for the coma coefficient a₃^-1 and the trefoil coefficient a₃^-3. Again, the computer model clearly mimics the statistical correlation present in the human eye data.

To verify the statistical correlation between all possible pairs of Zernike coefficients produced by the model, we computed the correlation matrix for the sample of 1,000 virtual eyes. The result, shown in Fig. 8, has all the features noted previously for human eyes (Fig. 3). A quantitative comparison of the correlation matrices in Figs. 3 and 8 was drawn by calculating the difference between corresponding correlation coefficients. The frequency distribution for these differences is shown as an inset in Fig. 8. The mean of this distribution is 0.0002 and the standard deviation is 0.03, which indicates very close agreement.

DISCUSSION

A variety of potential applications are likely to benefit from a statistical model of the eye’s WAF. For example, virtual eyes may be useful for computer simulation of individual variation in aberration structure, retinal image quality, visual performance, or response to optical treatment. In a clinical context, a population of virtual eyes may prove useful for predicting the fraction of individuals who will benefit from new designs of contact lenses, spectacles, or intra-ocular lenses. Given a model of how such ophthalmic devices interact with the eye to produce a total optical system, a given optical design might be tested against a large number of virtual eyes, thereby reducing the need for costly clinical trials. Similarly, new concepts for refractive surgery might be evaluated with virtual eyes to gain insight into the likely outcome prior to human experimentation.

Our statistical model eye may also be useful for evaluating visual function. For each virtual eye with hypothetical optical aberration, we may assess the quality of the retinal image by computing the corresponding point-spread function (PSF), optical transfer function (OTF), or retinal image for an arbitrary object. From each of these functions we may compute summary metrics of image quality, such as Strehl Ratio, width of the PSF, volume under the OTF, cutoff spatial frequency, or image contrast. Given these descriptions, and a model of visual function, we might predict visual performance on such tasks as visual acuity, contrast sensitivity, or target detection.

The statistical model eye presented here matches the statistical properties of a population of young, healthy, normal adult eyes. If other populations can be successfully modeled in a similar fashion, then the concept of a statistical model eye may provide a new approach to studying the differences between populations.Acknowledgements

The authors which to acknowledge the essential contributions made by the team of investigators who helped conduct the Indiana Aberration Study. This study group includes Prof. D. Miller; research optometrists C. Riley and N. Himebaugh, graduate student X. Cheng, optometry students E. Agapios T. Cao, S. Kaluf, J. McKenna, M. Price, P. Quach, J. Rajasansi, J. Tsai, and K. Vicari, and technical staff D. Carter, L. Wagoner, and K. Haggerty. Financial support was provided by NEI grants R01-EY05109 to LNT and NEI-STTR grant 1R41EY12754-01 to Quarrymen Optical, Inc.

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