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Journal ArticleDOI

Power vectors: An application of Fourier analysis to the description and statistical analysis of refractive error

01 Jun 1997-Optometry and Vision Science (Optom Vis Sci)-Vol. 74, Iss: 6, pp 367-375
TL;DR: 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.
Abstract: 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 degree and the other at axis 45 degrees. 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.
Citations
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Journal ArticleDOI
TL;DR: It is inferred that subjective best focus occurs when the area of the central, aberration-free region of the pupil is maximized, and 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.
Abstract: 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.

615 citations

Journal ArticleDOI
TL;DR: It is concluded that objective methods of refraction based on wavefront aberration maps can accurately predict the results of subjective refraction and may be more precise and wavefront methods may become the new gold standard for specifying conventional and/or optimal corrections of refractive errors.
Abstract: 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.

560 citations


Cites background from "Power vectors: An application of Fo..."

  • ...The idea of blur strength is to think of the wavefront locally as a small piece of a quadratic surface for which a power vector representation can be computed (Thibos et al., 1997)....

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Journal ArticleDOI
TL;DR: Use of progressive addition lenses compared with SVLs slowed the progression of myopia in COMET children by a small, statistically significant amount only during the first year, and provided some support for the COMET rationale, a role for defocus in progression ofMyopia.
Abstract: Purpose The purpose of the Correction of Myopia Evaluation Trial (COMET) was to evaluate the effect of progressive addition lenses (PALs) compared with single vision lenses (SVLs) on the progression of juvenile-onset myopia. Methods COMET enrolled 469 children (ages 6-11 years) with myopia between -1.25 and -4.50 D spherical equivalent. The children were recruited at four colleges of optometry in the United States and were ethnically diverse. They were randomly assigned to receive either PALs with a +2.00 addition (n = 235) or SVLs (n = 234), the conventional spectacle treatment for myopia, and were followed for 3 years. The primary outcome measure was progression of myopia, as determined by autorefraction after cycloplegia with 2 drops of 1% tropicamide at each annual visit. The secondary outcome measure was change in axial length of the eyes, as assessed by A-scan ultrasonography. Child-based analyses (i.e., the mean of the two eyes) were used. Results were adjusted for important covariates, by using multiple linear regression. Results Of the 469 children (mean age at baseline, 9.3 +/- 1.3 years), 462 (98.5%) completed the 3-year visit. Mean (+/-SE) 3-year increases in myopia (spherical equivalent) were -1.28 +/- 0.06 D in the PAL group and -1.48 +/- 0.06 D in the SVL group. The 3-year difference in progression of 0.20 +/- 0.08 D between the two groups was statistically significant (P = 0.004). The treatment effect was observed primarily in the first year. The number of prescription changes differed significantly by treatment group only in the first year. At 6 months, 17% of the PAL group versus 30% of the SVL group needed a prescription change (P = 0.0007), and, at 1 year, 43% of the PAL group versus 59% of the SVL group required a prescription change (P = 0.002). Interaction analyses identified a significantly larger treatment effect of PALs in children with lower versus higher baseline accommodative response at near (P = 0.03) and with lower versus higher baseline myopia (P = 0.04). Mean (+/- SE) increases in the axial length of eyes of children in the PAL and SVL groups, respectively, were: 0.64 +/- 0.02 mm and 0.75 +/- 0.02 mm, with a statistically significant 3-year mean difference of 0.11 +/- 0.03 mm (P = 0.0002). Mean changes in axial length correlated with those in refractive error (r = 0.86 for PAL and 0.89 for SVL). Conclusions Use of PALs compared with SVLs slowed the progression of myopia in COMET children by a small, statistically significant amount only during the first year. The size of the treatment effect remained similar and significant for the next 2 years. The results provide some support for the COMET rationale-that is, a role for defocus in progression of myopia. The small magnitude of the effect does not warrant a change in clinical practice.

550 citations


Cites methods from "Power vectors: An application of Fo..."

  • ...Progression of myopia was analyzed by expressing refractive error as three components: M (spherical equivalent), J0 (dioptric power of a Jackson cross cylinder at axis 0°), and J45 (dioptric power of a Jackson cross cylinder at axis 45°), as determined by Fourier decomposition.(16) Because oblique astigmatism is often mirror symmetric in the two eyes, the average J45 values were calculated by transforming the axis values between 91° and 180° to values between 0° and 90° for each eye and then averaging them between the two eyes....

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Journal ArticleDOI
TL;DR: Topical atropine was well tolerated and effective in slowing the progression of low and moderate myopia and ocular axial elongation in Asian children.

462 citations


Cites methods from "Power vectors: An application of Fo..."

  • ...Progression of myopia was analyzed by expressing refractive error as 3 components: M (spherical equivalent), J0 (dioptric power of a Jackson cross cylinder at axis 0), and J45 (dioptric power of a Jackson cross cylinder at axis 45), as determined by Fourier decomposition.(19) The secondary outcome was change in axial length during follow-up relative to baseline measured by A-scan ultrasonography....

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Journal ArticleDOI
TL;DR: Frequency distributions of blur strength clearly demonstrate the effectiveness of refractive surgery in reducing the overall blurring effect of uncorrected refractive error and power vector analysis revealed a reduction in the astigmatic component of these refractive errors.
Abstract: 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 J 0 with axes at 90 degrees and 180 degrees, and a Jackson crossed cylinder of power J 45 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.

426 citations


Cites background from "Power vectors: An application of Fo..."

  • ...0 for 1cyl form, C , 0 for 2cyl form).(12)...

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  • ...Power vectors are a geometrical representation of spherocylindrical refractive errors in 3 fundamental dioptric components.(12) The first component is a spherical lens with power M equal to the spherical equivalent of the given refractive error....

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