Larry N. Thibos

Power Vectors: An Application of Fourier Analysis to the Description and Statistical Analysis of Refractive Error

The description of sphero-cylinder lenses is approached from the viewpoint of Fourier analysis of the power profile. It is shown that the familiar sine-squared law leads naturally to a Fourier series representation with exactly three Fourier coefficients, representing the natural parameters of a thin lens. The constant term corresponds to the mean spherical equivalent (MSE) power, whereas the amplitude and phase of the harmonic correspond to the power and axis of a Jackson cross-cylinder (JCC) lens, respectively. Expressing the Fourier series in rectangular form leads to the representation of an arbitrary sphero-cylinder lens as the sum of a spherical lens and two cross-cylinders, one at axis 0° and the other at axis 45°. The power of these three component lenses may be interpreted as (x,y,z) coordinates of a vector representation of the power profile. Advantages of this power vector representation of a sphero-cylinder lens for numerical and graphical analysis of optometric data are described for problems involving lens combinations, comparison of different lenses, and the statistical distribution of refractive errors.

Statistical variation of aberration structure and image quality in a normal population of healthy eyes

A Shack-Hartmann aberrometer was used to measure the monochromatic aberration structure along the primary line of sight of 200 cyclopleged, normal, healthy eyes from 100 individuals. Sphero-cylindrical refractive errors were corrected with ophthalmic spectacle lenses based on the results of a subjective refraction performed immediately prior to experimentation. Zernike expansions of the experimental wave-front aberration functions were used to determine aberration coefficients for a series of pupil diameters. The residual Zernike coefficients for defocus were not zero but varied systematically with pupil diameter and with the Zernike coefficient for spherical aberration in a way that maximizes visual acuity. We infer from these results that subjective best focus occurs when the area of the central, aberration-free region of the pupil is maximized. We found that the population averages of Zernike coefficients were nearly zero for all of the higher-order modes except spherical aberration. This result indicates that a hypothetical average eye representing the central tendency of the population is nearly free of aberrations, suggesting the possible influence of an emmetropization process or evolutionary pressure. However, for any individual eye the aberration coefficients were rarely zero for any Zernike mode. To first approximation, wave-front error fell exponentially with Zernike order and increased linearly with pupil area. On average, the total wave-front variance produced by higher-order aberrations was less than the wave-front variance of residual defocus and astigmatism. For example, the average amount of higher-order aberrations present for a 7.5-mm pupil was equivalent to the wave-front error produced by less than 1/4 diopter (D) of defocus. The largest pupil for which an eye may be considered diffraction-limited was 1.22 mm on average. Correlation of aberrations from the left and right eyes indicated the presence of significant bilateral symmetry. No evidence was found of a universal anatomical feature responsible for third-order optical aberrations. Using the Marechal criterion, we conclude that correction of the 12 largest principal components, or 14 largest Zernike modes, would be required to achieve diffraction-limited performance on average for a 6-mm pupil. Different methods of computing population averages provided upper and lower limits to the mean optical transfer function and mean point-spread function for our population of eyes.

Accuracy and precision of objective refraction from wavefront aberrations

We determined the accuracy and precision of 33 objective methods for predicting the results of conventional, sphero-cylindrical refraction from wavefront aberrations in a large population of 200 eyes. Accuracy for predicting defocus (as specified by the population mean error of prediction) varied from -0.50 D to +0.25 D across methods. Precision of these estimates (as specified by 95% limits of agreement) ranged from 0.5 to 1.0 D. All methods except one accurately predicted astigmatism to within +/-1/8D. Precision of astigmatism predictions was typically better than precision for predicting defocus and many methods were better than 0.5D. Paraxial curvature matching of the wavefront aberration map was the most accurate method for determining the spherical equivalent error whereas least-squares fitting of the wavefront was one of the least accurate methods. We argue that this result was obtained because curvature matching is a biased method that successfully predicts the biased endpoint stipulated by conventional refractions. Five methods emerged as reasonably accurate and among the most precise. Three of these were based on pupil plane metrics and two were based on image plane metrics. We argue that the accuracy of all methods might be improved by correcting for the systematic bias reported in this study. However, caution is advised because some tasks, including conventional refraction of defocus, require a biased metric whereas other tasks, such as refraction of astigmatism, are unbiased. We conclude that objective methods of refraction based on wavefront aberration maps can accurately predict the results of subjective refraction and may be more precise. If objective refractions are more precise than subjective refractions, then wavefront methods may become the new gold standard for specifying conventional and/or optimal corrections of refractive errors.

Power vector analysis of the optical outcome of refractive surgery

Purpose: To demonstrate the power vector method of representing and analyzing spherocylindrical refractive errors. Setting: School of Optometry, Indiana University, Bloomington, Indiana, USA. Methods: Manifest and keratometric refractive errors were expressed as power vectors suitable for plotting as points in a 3‐dimensional dioptric space. The 3 Cartesian coordinates (x, y, z) of each power vector correspond to the powers of 3 lenses that, in combination, fulfill a refractive prescription: a spherical lens of power M, a Jackson crossed cylinder of power J0 with axes at 90 degrees and 180 degrees, and a Jackson crossed cylinder of power J45 with axes at 45 degrees and 135 degrees. The Pythagorean length of the power vector, B, is a measure of overall blurring strength of a spherocylindrical lens or refractive error. Changes in refractive error due to surgery were computed by the ordinary rules of vector subtraction. Results: Frequency distributions of blur strength (B) clearly demonstrate the effectiveness of refractive surgery in reducing the overall blurring effect of uncorrected refractive error. Power vector analysis also revealed a reduction in the astigmatic component of these refractive errors. Paired comparisons revealed that the change in manifest astigmatism due to surgery was well correlated with the change in keratometric astigmatism. Conclusions: Power vectors aid the visualization of complex changes in refractive error by tracing a trajectory in a uniform dioptric space. The Cartesian components of a power vector are mutually independent, which simplifies mathematical and statistical analysis of refractive errors. Power vectors also provide a natural link to a more comprehensive optical description of ocular refractive imperfections in terms of wavefront aberration functions and their description by Zernike polynomials.

Metrics of optical quality derived from wave aberrations predict visual performance

Wavefront-guided refractive surgery and custom optical corrections have reduced the residual root mean squared (RMS) wavefront error in the eye to relatively low levels (typically on the order of 0.25 microm or less over a 6-mm pupil, a dioptric equivalent of 0.19 D). It has been shown that experimental variation of the distribution of 0.25 microm of wavefront error across the pupil can cause variation in visual acuity of two lines on a standard logMAR acuity chart. This result demonstrates the need for single-value metrics other than RMS wavefront error to quantify the effects of low levels of aberration on acuity. In this work, we present the correlation of 31 single-value metrics of optical quality to high-contrast visual acuity for 34 conditions where the RMS wavefront error was equal to 0.25 microm over a 6-mm pupil. The best metric, called the visual Strehl ratio, accounts for 81% of the variance in high-contrast logMAR acuity.

Clinical applications of the Shack-Hartmann aberrometer.

The efficacy of the Shack-Hartmann technique for measuring the optical aberrations of the eye was evaluated for four classes of clinical conditions associated with optically abnormal eyes. These categories (with specific examples) are: anomalies of the tear film (dry eye), corneal disease (keratoconus), corneal refractive surgery [laser-assisted in situ keratomileusis (LASIK)], and lenticular cataract. We show that in each of these cases, it is possible to obtain at least a partial topographic map of the refractive aberrations of the patients eyes, but severe losses of data integrity can occur. We further show that the Shack-Hartmann aberrometer provides additional information about the eyes imperfections on a very fine spatial scale (< 0.4 mm) which scatter light and further degrade the quality of the retinal image. Taken together, spatial maps of the variation of optical aberrations and scatter across the eyes entrance pupil represents an improved description of the optical imperfections of the abnormal eye.

The chromatic eye: a new reduced-eye model of ocular chromatic aberration in humans

New measurements of the chromatic difference of focus of the human eye were obtained with a two-color, vernier-alignment technique. The results were used to redefine the variation of refractive index of the reduced eye over the visible spectrum. The reduced eye was further modified by changing the refracting surface to an aspherical shape to reduce the amount of spherical aberration. The resulting chromatic-eye model provides an improved account of both the longitudinal and transverse forms of ocular chromatic aberration.

Theory and measurement of ocular chromatic aberration

We have determined the transverse chromatic aberration of the human eye by measuring the apparent offset of a two-color vernier viewed foveally through a displaced, pinhole aperture. For the same subjects, we also determined the longitudinal chromatic aberration for foveal viewing by the method of best focus. In both cases, the results were closely predicted by a simple, reduced-eye optical-model for which transverse and longitudinal chromatic aberration are directly proportional, with the constant of proportionally being the amount of displacement of the pinhole from the visual axis. Further measurements revealed that the natural pupil was closely centered on the visual axis for two subjects and slightly displaced in the temporal direction for three other subjects. One implication of these results is that, although the eye has substantial chromatic aberration, the pupil is positioned so as to minimize the transverse component of the aberration for central vision, thereby optimizing foveal image quality for polychromatic objects.

Predicting subjective judgment of best focus with objective image quality metrics

PURPOSE To determine the impact of higher-order monochromatic aberrations on lower-order subjective sphero-cylindrical refractions. METHODS Computationally-aberrated, monochromatic Sloan letters were presented on a high luminance display that was viewed by an observer through a 2.5mm pupil. Through-focus visual acuity (VA) was determined in the presence of spherical aberration (Z40) at three levels (0.10, 0.21 and 0.50D). Analogous through-astigmatism experiments measured visual acuity in the presence of secondary astigmatism (Z4+/-2) or coma (Z3-1). Measured visual acuity was correlated with 31 different metrics of image quality to determine which metric best predicts performance for degraded retinal images. The defocus and astigmatism levels that optimized each metric were compared with those that produced best visual acuity to determine which metric best predicts subjective refraction. RESULTS Spherical aberration, coma and secondary astigmatism all reduced VA and increased depth of focus. The levels of defocus and primary astigmatism that produced the best performance varied with levels of spherical aberration and secondary astigmatism, respectively. The presence of coma, however, did not affect cylindrical refraction. Image plane metrics, especially those that take into account the neural contrast sensitivity threshold (e.g. the visual Strehl ratio, VSOTF), are good predictors of visual acuity in both the through-focus and through-astigmatism experiments (R = -0.822 for VSOTF). Subjective sphero-cylindrical refractions were accurately predicted by some image-quality metrics (e.g., pupil fraction, VSOTF and standard deviation of PSF light distribution). CONCLUSION Subjective judgment of best focus does not minimize RMS wavefront error (Zernike defocus = 0), nor create paraxial focus (Seidel defocus = 0), but makes the retina conjugate to a plane between these two. It is possible to precisely predict subjective sphero-cylindrical refraction for monochromatic light using objective metrics.

RELATIONSHIP BETWEEN REFRACTIVE ERROR AND MONOCHROMATIC ABERRATIONS OF THE EYE

Purpose. To examine the relationship between ametropia and optical aberrations in a population of 200 normal human eyes with refractive errors spanning the range from +5.00 to −10.00 D. Methods. Using a reduced-eye model of ametropia, we tested the hypothesis that the optical system of the eye is uncorrelated with the degree of ametropia. These predictions were evaluated experimentally with a Shack-Hartmann aberrometer that measured the monochromatic aberrations across the central 6 mm of the dilated pupil in well-corrected, cyclopleged eyes. Results. Optical theory predicted, and control experiments on a model eye verified, that Shack-Hartmann measurements of spherical aberration will vary with axial elongation of the eye even if the dioptric components of the eye are fixed. Contrary to these predictions, spherical aberration was not significantly different from emmetropic eyes. Root mean square of third-order aberrations, fourth-order aberrations, and total higher aberrations (third to 10th) in myopic and hyperopic eyes were also uncorrelated with refractive error. Astigmatic eyes tended to have larger total higher-order aberrations than nonastigmatic eyes. Conclusions. We conclude that a reduced-eye model of myopia assuming fixed optical parameters and variable axial length is not tenable.