Unbiased Estimation of Refractive State of Aberrated Eyes

Abstract

To identify unbiased methods for estimating the target vergence required to maximize visual acuity based on wavefront aberration measurements. Experiments were designed to minimize the impact of confounding factors that have hampered previous research. Objective wavefront refractions and subjective acuity refractions were obtained for the same monochromatic wavelength. Accommodation and pupil fluctuations were eliminated by cycloplegia. Unbiased subjective refractions that maximize visual acuity for high contrast letters were performed with a computer controlled forced choice staircase procedure, using 0.125 diopter steps of defocus. All experiments were performed for two pupil diameters (3mm and 6mm). As reported in the literature, subjective refractive error does not change appreciably when the pupil dilates. For 3 mm pupils most metrics yielded objective refractions that were about 0.1D more hyperopic than subjective acuity refractions. When pupil diameter increased to 6 mm, this bias changed in the myopic direction and the variability between metrics also increased. These inaccuracies were small compared to the precision of the measurements, which implies that most metrics provided unbiased estimates of refractive state for medium and large pupils. A variety of image quality metrics may be used to determine ocular refractive state for monochromatic (635nm) light, thereby achieving accurate results without the need for empirical correction factors.

INTRODUCTION

Accurate and precise measurement of the eye’s refractive state is fundamental to many aspects of vision science and clinical practice. Clinicians typically measure refractive state of the un-accommodated eye to determine prescriptions for spectacles, contact lenses, or refractive surgery. Studies of accommodation require measurement of the change in refractive state. When refractive state is inappropriate for the vergence of a visual target, the resulting blur affects visual performance and may induce growth changes leading to myopia. For all of these applications the paraxial measurement of refractive state is conceptually and technologically straightforward but the results do not take account of the effects of higher-order aberrations in non-paraxial regions of the pupil.

Refractive state can be measured by subjective or objective methods, so it is important to define refractive state in a way that is compatible with both classes of measurement. In this report we define refractive state as the target vergence at the corneal plane required to maximize retinal image quality (Lopez-Gil, Fernandez-Sanchez, Thibos & Montes-Mico, 2009). This definition is compatible with subjective methodologies (e.g. clinical refraction) that seek to maximize performance on a spatial vision task, provided we adopt the fundamental assumption that retinal image quality is maximized when visual performance is maximized (Ravikumar, Thibos & Bradley, 2008). Our definition is also compatible with objective methodologies such as autorefractors that determine refractive state by maximizing the contrast of the double-pass aerial-image of a test pattern formed on the fundus (Campbell, Benjamin & Howland, 2006). However, the principles of operation for most commercial autorefractors (e.g. Scheiner’s disk, retinoscopy, knife-edge, or ray deflection) are based on techniques that are only indirectly linked to image quality.

Modern wavefront aberrometers can be used to determine refractive state, as defined above, by computing retinal image quality from measurements of wavefront error. The Shack-Hartmann wavefront aberrometer, for example, yields a comprehensive description of the eye’s optical aberrations, from which the retinal image can be computed using the theories of physical or geometrical optics (Hopkins, 1950). From such computations refractive state can be estimated by determining the spherical wavefront that, when added to the measured wavefront, will maximize retinal image quality. Retinal image quality can be quantified for this purpose by a variety of metrics, some of which exhibit significant correlation with visual performance (Cheng, Bradley & Thibos, 2004 2004). This objective process of wavefront refraction offers an opportunity to determine the refractive state of the eye in any state of accommodation that is consistent with subjective determinations (Guirao & Williams, 2003, Navarro, 2009, Thibos, Hong, Bradley & Applegate, 2004).

Many factors must be taken into account to achieve unbiased estimates of refractive state from wavefront measurements. (1) Aberrometers use reflected light rather than the visually-relevant light absorbed by photoreceptors. (2) Wavefront measurements are based on monochromatic light whereas the “gold standard” of subjective refraction is typically determined with polychromatic light. (3) Refractive power of the eye for infrared light commonly used in aberrometers is less than for visible wavelengths (Llorente, Diaz-Santana, Lara-Saucedo & Marcos, 2003). (4) The axial location of reflected light may be different from the entrance apertures of cone photoreceptors (Gao, Cense, Zhang, Jonnal & Miller, 2008, Gao, Jonnal, Cense, Kocaoglu, Wang & Miller, 2009) by an amount that varies with wavelength (Elsner, Burns, Weiter & Delori, 1996). (5) Pupil sizes for objective and subjective refractions must match, and appropriate weighting of the pupil function may be required, to take into account factors such as the Stiles-Crawford effect when computing image quality from aberrometry data. (6) Inappropriate accommodation (i.e. “instrument myopia”) produces a misleading estimate of refractive state for the relaxed eye. (7) Wavefront refractions are typically referenced to the eye’s entrance pupil plane, whereas subjective refractions are referenced to either the cornea, or the spectacle plane, or the phoropter lens plane thereby introducing discrepancies due to effectivity. (8) Even if an objective refraction methodology is unbiased, it may appear biased when compared against a biased subjective refraction. For example, the “maximum plus” technique used in clinical refraction (Borisch, 1970) intentionally prescribes spherical power to focus light from the hyperfocal distance (Campbell, 1957) rather than infinity (the reference distance in wavefront refraction). Our study was designed to minimize the impact of these numerous sources of potential bias in both objective and subjective refractive measures to determine if any metric of image quality leads to unbiased measurements of refractive state.

1. METHODS

1.1. Subjects

Eighteen young adult subjects (ages 23–36 yr) and two older adults (54–59 yr) with low refractive errors (mean spherical equivalent −4.75 to +1.25 D) were recruited from the Indiana University School of Optometry. Experimental protocols were approved by the Indiana University Institutional Review Board and all subjects gave informed consent. Prior to cycloplegia, intraocular pressure and anterior chamber angles were evaluated and eye clinic records were examined. All subjects had best-corrected visual-acuity better than 20/20 with no signs of ocular pathology.

Cycloplegia was induced in right eyes by instilling one drop of 1% cyclopentolate. Aberrometry was performed 25–45 minutes later with a COAS aberrometer (Wavefront Sciences, Inc.) modified to use visible light (635nm). After 30 minutes, a nominal subjective refraction was performed in the laboratory while viewing 20/20 and 20/15 black letters on a white background through the same telescope used for experiments. Subjects indicated which combination of spherical and cylindrical ophthalmic lenses provided maximum clarity of vision. The mean spherical equivalent of this optimum correcting lens was taken as the zero-defocus value in the thru-focus experiment. Starting from this nominally well-corrected state, we determined the power (plus or minus) of an additional spherical lens needed to optimize acuity for monochromatic targets. Refractive errors reported in Results refer to the power of this additional defocusing lens which, in clinical parlance, would be called an over-refraction.

Wavefront aberrations of each subject’s dilated right eye were measured for 635nm light using the modified aberrometer described below. Aberration measurements obtained for dilated pupils were resized to 6mm or 3mm pupil diameters by applying a mask to the raw data image and then fitting the masked data with derivatives of Zernike polynomials. Spherical aberration was positive for 17 of 20 subjects. Figure 1 shows population mean values and standard deviations of higher-order RMS wavefront error for Zernike orders 3–8 and the total of those orders for both pupil sizes. These values are typical of normal, healthy, young adult subjects (Thibos, Hong, Bradley & Cheng, 2002).

Figure 1.

Higher order aberrations of subject population for 635nm light. Symbols indicate population means and error bars indicate ±1 standard deviation of the population.

1.2. Wavefront aberrometry

A commercially available COAS aberrometer (Wavefront Sciences, Inc.) was converted to measure the aberrations of the human eye at the visible wavelength of 635 nm. The internal infrared (840 nm) source path was blocked and an external probe beam from a semiconductor laser (635 nm) was injected into the eye by a beam splitter between COAS and the eye. This modification required changing the first beam splitter inside COAS so that light reflected by the fundus from the 635nm probe beam would enter the wavefront sensor channel of the aberrometer. The new beam splitter (Chroma Technology Corp) had a reflection of 95% at 635 nm. Following modifications, a new reference file was created for the visible 635nm source. Factory calibration of the modified aberrometer was verified by introducing into the aberrometer path spherical wavefronts (−4D to +4D) produced by a laser beam defocused with corrected-curve ophthalmic trial lenses. The measured defocus (y) was linearly correlated with wavefront vergence (x) (R² = 0.99, Y=0.98x−0.02) which confirmed the instrument’s ability to accurately measure spherical wavefront errors at 635 nm.

Further validation of the modified instrument compared the spherical refraction between the external 635nm source and the internal 850nm source measured with a model eye exhibiting known chromatic aberration. The model eye contained a plano-convex lens (f=25mm, diameter=12.7, BK7 glass) and a white paper retina at its focal length with a neutral density 0.9 filter placed before the lens to act as density 1.8 in double pass. This configuration provided similar reflectivity (10⁻⁴) found for human eyes with visible light. The chromatic aberration of the model eye was correctly measured to within 0.02D of the expected value (0.55D). The higher order aberrations of this model eye measured with the modified COAS were within 0.01 microns (trefoil) and 0.04 (spherical aberration) of those measured on two other unmodified COAS instruments.

1.3. Subjective acuity refraction

In this report, the term “acuity refraction” refers to the spherical power of an ophthalmic correcting lens needed to maximize visual acuity. Visual performance was determined using the same portion of the eye’s optics and the same wavelength of light (635 nm) used to measure the eye’s monochromatic aberrations. Subjects viewed letter targets through a unit magnification telescope that imaged an aperture (3 or 6 mm diameter) into the pupil plane of the subject’s eye. The eye was aligned to the instrument axially and transversely with the aid of a pupil alignment CCD camera and an alignment target (Fig. 2). The alignment target was concentric with the optical axis of the viewing channel and conjugate to the artificial pupil and to the eye’s entrance pupil. Spherical and cylindrical trial lenses were placed in a lens holder 17mm anterior to the telescope entrance aperture. Spherical lens power was changed in 1/8 diopter increments to introduce a counterbalanced series of positive and negative defocus relative to the nominal, well-focused condition determined initially for white light. Alignment was maintained with a dental bite bar to stabilize the head and was monitored throughout the experiments.

Figure 2.

Experimental apparatus for measuring visual acuity as a function of defocus. An afocal telescope with unit magnification (Lens 1, Lens 2) imaged the artificial pupil (3mm or 6mm diameter) into the plane of the eye’s entrance pupil. Lens 3 conjugated the alignment target to the eye’s entrance pupil as seen by the alignment camera.

Size, contrast, timing and position of test letters were controlled by customized MatLab (Mathworks, MA) programs running on an IMac computer (Apple Corp., CA). Letter targets were randomly selected from the Sloan letter set (C,D,H,K,N,O,R,S,V,Z). A digital projector (Dell 7609WU) and rear projection screen was used to generate a high resolution (1920 × 1200 pixels) and high luminance (3000 lux) gamma-corrected white display. This approach allowed the subject to view the display through an interference filter (peak transmission at 635 nm, full-width at half-height = 17 nm) while retaining high-photopic light levels (250 cd/m²). White test stimuli of the same mean luminance were generated by viewing the display through a 1.0 ND neutral density filter.

Visual acuity (VA) experiments presented a randomized sequence of letters in a staircase paradigm. Each trial contained a 1 second stimulus presentation signaled by a tone. Subsequent presentations were triggered by the subject’s response entered on a 10-number keypad. A short training session enabled subjects to use a number keypad to code each response. No feedback was given. Letter size was determined by a staircase procedure (Levitt, 1971), with initial and final size steps of 0.2 and 0.05 logMAR respectively. Letter size was increased following an incorrect report, and decreased following 2 successive correct responses providing a 71% correct asymptote. Threshold letter size was determined as the mean of the last 10 of 12 reversals (in logMAR units). The standard deviation of the last 10 reversals was used as a measure of precision of this VA value. The power of the added spherical lens that optimized visual was designated the “acuity refraction”.

1.4. Objective wavefront refraction

The Zernike aberration coefficient C₂⁰ for defocus obtained for the naked eye was corrected for target viewing distance (2.5 m), and for the ophthalmic lenses used in the apparatus. This enabled a direct comparison of objective refractions derived from the corrected wavefront with subjective refractions obtained at the same target distance. Objective refractions were determined by two strategies described in detail previously (Thibos et al., 2004). The first strategy finds the best fitting (i.e. “equivalent”) sphere to represent the measured ocular wavefront. Fitting can be accomplished by two criteria designated here “Zernike” and “Seidel”. The Zernike refraction minimizes the RMS (root-mean-squared) deviation between the ocular wavefront and the fitted sphere. This method reports refractive state by the defocus coefficient C₂⁰ in a Zernike expansion of the ocular wavefront (converted to diopters using eqn. (1) in (Thibos et al., 2004)). Thus a Zernike refraction specifies the vergence of a point source that focuses a “disk of least confusion” into the plane of reflection of the aberrometer’s probe beam. Alternatively, a Seidel refraction matches the meridionally-averaged paraxial curvature of the ocular wavefront to the curvature of the fitted sphere. This result reports refractive state by the r² coefficient of a power-series expansion of the ocular wavefront (converted to diopters using eqn. (2) in (Thibos et al., 2004)). A Seidel refraction specifies the vergence of a point source that focuses paraxial rays into the plane of reflection of the aberrometer’s probe beam. In the absence of higher-order aberrations, Zernike and Seidel refractions are identical.

The second strategy for wavefront refraction finds that spherical wavefront which, when added to the ocular wavefront, optimizes retinal image quality. This was achieved computationally by adding to the ocular wavefront a series of defocused wavefronts that simulate the defocus lenses employed during the visual acuity experiment. For each of these defocused ocular wavefronts we computed image quality using 31 objective metrics described previously (Thibos et al., 2004). The amount of defocus needed to optimize each metric of image quality was designated the objective refraction for that metric. Thus with 31 metrics, plus the Zernike and Seidel refractions described above, we achieved 33 objective refractions from each eye’s aberrometry data to be compared directly to the subjective acuity refractions obtained in the psychophysical experiment.

In order to compute image quality for each state of defocus, it was necessary to use the wavefront error map to compute the point-spread function (PSF), and the optical transfer function (OTF). The PSF was convolved with an eye chart to simulate the retinal image, from which RMS contrast was computed as the standard deviation of pixel values in the image. This RMS contrast served as the 34^th metric for objective refraction. All optical calculations were performed in Matlab using custom programs.

2. RESULTS

An example of a through-focus curve used to determine acuity refraction is shown in Fig. 3 for a medium (3mm) diameter pupil. The term “acuity refraction” refers to the abscissa value for which logMAR is lowest (i.e. best acuity). Since every point on the curve is subject to measurement variance, from a statistical viewpoint more than one point on the curve could legitimately be considered the minimum. To select a unique lens value, we adopted the following procedure. All points that had confidence intervals that overlap with the confidence interval for the lowest point on the curve were considered equally valid candidates (filled symbols in Fig. 3). These form a trough in the curve defining the eye’s depth-of-field. To avoid the inherent bias of the “maximum plus” criterion used clinically, we selected the median abscissa values for these trough points as the acuity refraction. The median of the corresponding ordinate values was designated as the value of acuity achieved under best-focus conditions. This median point representing the minimum of the curve, indicated graphically by an open triangle in Fig. 3, was the endpoint of our acuity refraction procedure. The dioptric range of the trough points was taken as a measure of the magnitude of the depth-of-field.

Figure 3.

An example of acuity and objective refractions. Open circles represent visual acuity (left-hand ordinate) obtained for the indicated level of defocus (abscissa). Error bars indicate a confidence interval (±2 sem) for the mean of 10 staircase reversals. Acuity values not statistically different from the lowest point form a trough indicated by filled circles. Median abscissa value of the trough points (shown by the open triangle) was taken as the subjective refraction endpoint. Objective refraction of this eye using −log(area-under-the-MTF) as a metric of image degradation is given by the low point on the curve formed by plus (+) symbols in reference to the right-hand ordinate. The lateral displacement of the minima of these two curves represents a discrepancy between objective and subjective refractions. Both logarithmic ordinates span 1 log unit, but otherwise are unrelated.

Figure 3 also illustrates the corresponding procedure for objective refraction using a specific example of a metric of image quality (area under the MTF). To enhance visual comparison with logMAR acuity, Fig. 3 displays image degradation (the inverse of area under MTF) plotted on a logarithmic axis. These calculated curves of image quality tend to be very smooth with a well-defined minimum that was adopted as the endpoint of the objective refraction procedure. The lateral displacement of the objective minimum from the subjective minimum in this example indicates a hyperopic bias of the metric-based objective refraction relative to the subjective refraction. Since each metric emphasizes a different aspect of image quality, the amount of bias was different for every metric and for every subject but nevertheless trends emerged that are described below.

2.1. Subjective refractions

Using the method illustrated in Fig. 3, we measured acuity refractions for 20 eyes and the results for medium and large pupils are shown in Fig. 4. The population mean for the 3mm pupil (+0.19 D) was not significantly different from the population mean for the 6mm pupil (+0.23D), and the slope of the orthogonal regression line (1.05) was not significantly different from unity. These results confirm earlier findings (Charman, Jennings & Whitefoot, 1978, Koomen, Tousey & Scolnik, 1949) that refractive error does not change appreciably when the pupil dilates. Each of the data points in Fig. 4 lies at the center of a range of dioptric values in both x and y representing the eye’s depth of field. The average range for 3mm pupils was 0.56 D and for 6mm pupils was 0.48D. For 17 of 20 subjects one or both of these ranges overlapped the regression line, indicating the subjective refractions for the two pupil sizes were not functionally different.

Figure 4.

Comparison of subjective acuity refraction for medium (3mm) and large (6mm) pupils for 635nm light. Abscissa and ordinate values are in diopters, relative to refraction for white light. Open circles indicate the minimum of the through-focus curves for individual subjects as determined by the method illustrated in Fig. 3. Filled circle indicates the population mean of abscissa and ordinate values, for which the 95% confidence region is shown by the ellipse. Filled triangle indicates the predicted mean value based on the Indiana Eye model of ocular chromatic aberration. The orthogonal regression line is also the first principal component of the data. The average depth of focus for 3mm pupils was 0.56 D and for 6mm pupils was 0.48D.

The positive values of the population mean refractions for both pupil sizes were attributed to ocular chromatic aberration, which causes the eye to be slightly hyperopic for 635 nm when the eye is well focused for white light (recall that zero on the abscissa indicates the initial subjective refraction for white light). The anticipated mean value of 0.29D, predicted by the Indiana Eye model of chromatic aberration (Thibos, Ye, Zhang & Bradley, 1992) assuming the wavelength in focus for white light was 565nm (Coe, 2009), lies inside the confidence ellipse. Thus there is no basis for rejecting the hypothesis that ocular refractive chromatic aberration is responsible for the population mean refraction relative to white light refraction. The two refraction values were only weakly correlated (R=0.41, p=0.07), suggesting that the distribution of results about the mean is primarily due to random factors.

2.2. Objective Refraction

Results of objective refraction are shown in Fig. 5 for all 34 metrics (see Appendix for symbol key). All of the symbols except “2” (representing the peak-to-valley metric, see Appendix) lie to the right of the dotted vertical line redrawn from Fig. 4 for reference. This result indicates that, for 3 mm pupil diameters, the population mean of objective refractions was hyperopic relative to the subjective mean refraction for all metrics except one. To the contrary, the majority of points lie below the horizontal dashed line, indicating that for 6mm pupils the population mean of objective refractions were myopic relative to the subjective mean refraction. The metrics used in this study are known to be mutually correlated (Thibos et al., 2004), which accounts for the large degree of overlap of individual points in Fig. 5. Nevertheless, the dioptric range of refractions spanned by the various metrics was approximately threefold larger for 6mm pupils than for 3mm pupils. In the absence of higher order aberrations, all 34 objective metrics yield the same refraction, which implies that the differences between metrics evident in Fig. 5 are due to the variable impact of higher order aberrations on image quality as quantified by the various metrics. For a few metrics, the population mean of objective refraction lies just inside the confidence ellipse for the mean subjective refraction. Therefore, the statistical significance of the difference between subjective and objective mean refractions for these few metrics is marginal.

Figure 5.

Comparison of objective refractions for medium (3mm) and large (6mm) pupils for 635m light. Abscissa and ordinate values are in diopters, relative to refraction for white light. Unique symbols indicate the population mean objective refraction for each of the 34 metrics specified in Appendix. The population mean subjective refraction (filled circle), confidence ellipse for the mean, and regression line are repeated from Fig. 4 for reference.

APPENDIX.

The metrics of image quality used in this study

Plotting symbol	Acronym	Description
‘1’	‘RMSw’	root-mean-squared wavefront error computed over the whole pupil
‘2’	‘PV’	peak-to-valley difference in wavefront error
‘3’	‘RMSs’	root-mean-squared wavefront slope computed over the whole pupil
‘4’	‘PFWc’	pupil fraction for wavefront error, concentric pupil
‘5’	‘PFWt’	pupil fraction for wavefront error, tesselated pupil
‘6’	‘PFSt’	pupil fraction for wavefront slope, concentric pupil
‘7’	‘PFSc’	pupil fraction for wavefront slope, tesselated pupil
‘8’	‘Bave’	average blur strength
‘9’	‘PFCt’	pupil fraction for wavefront curvature, concentric pupil
‘A’	‘PFCc’	pupil fraction for wavefront curvature, tesselated pupil
‘B’	‘D50’	diameter of a circular area capturing 50% of the light energy
‘C’	‘EW’	equivalent width of centered PSF
‘D’	‘SM’	square root of second moment of light distribution
‘E’	‘HWHH’	half width at half height
‘F’	‘CW’	correlation width of light distribution in PSF
‘G’	‘SRX’	Strehl ratio computed in spatial domain
‘H’	‘LIB’	light-in-the-bucket
‘I’	‘STD’	standard deviation of intensity values in the PSF
‘J’	‘ENT’	entropy of the PSF
‘K’	‘NS’	Neural sharpness
‘L’	‘VSX’	visual Strehl ratio computed in the spatial domain
‘M’	‘SFcMTF’	spatial frequency cutoff of radially-averaged modulation-transfer function (rMTF)
‘N’	‘AreaMTF’	area of visibility for rMTF
‘O’	‘SFcOTF’	spatial frequency cutoff of radially-averaged optical-transfer function (rOTF)
‘P’	‘AreaOTF’	area of visibility for rOTF
‘Q’	‘SROTF’	Strehl ratio computed in frequency domain (OTF method)
‘R’	‘VOTF’	volume under OTF normalized by the volume under MTF
‘S’	‘VSOTF’	visual Strehl ratio computed in frequency domain (OTF method)
‘T’	‘VNOTF’	volume under neurally-weighted OTF, normalized by the volume under neurally weighted MTF
‘U’	‘SRMTF’	Strehl ratio computed in frequency domain (MTF method)
‘V’	‘VSMTF’	visual Strehl ratio computed in frequency domain (MTF method)
‘W’	‘Zernike’	optimum focusing of circle of least confusion
‘X’	Seidel	optimum focusing of paraxial rays
‘Y’	Contrast	rms contrast of pixels in a computed images of an eye chart

2.3. Comparison of individual subjective and objective refractions

The results presented above refer to population averages, but the efficacy of objective refraction in the clinic is measured by how well the objective refraction matches subjective refraction for each individual eye. Accordingly, we computed the difference between objective and subjective refraction for each individual eye for each of the 34 objective metrics. We measured inaccuracy by the population mean of those individual differences and we measured imprecision by the population standard deviation of those individual differences. Together the inaccuracy and imprecision of all 35 objective refractions for 3mm pupils define a cloud of points in Fig. 6 with each point marked by a unique symbol (see Appendix for symbol key). For most metrics, inaccuracy is positive, which means the objective refraction is more hyperopic than the subjective refraction, as anticipated by population mean data of Fig. 5.

Figure 6.

Efficacy of objective refraction for 3mm pupil diameter. Symbols show the inaccuracy and imprecision (in diopters) of objective refraction relative to subjective refraction. Inaccuracy is defined as the population mean of individual differences between objective and subjective refraction. Imprecision is defined as the standard deviation of the population of differences. Dashed lines define an acceptance region for the null hypothesis that inaccuracy = 0, which is equivalent to saying the metric is an unbiased estimator of subjective refraction. Only two metrics (A=PFCc and X=Seidel paraxial refraction) are biased by this test when pupil diameter is 3mm. Abscissa and ordinate values are in diopters.

To determine a threshold for inaccuracy that is statistically significant, we computed a confidence interval for the null hypothesis that inaccuracy = 0. The size of this confidence interval is ±2*precision/√N, where N=20 is the population size. Thus for every ordinate value in Fig. 6 we can erect a confidence interval centered on the abscissa origin and of width (4/√20)*ordinate value (diopters). The dashed lines connect the ends of these confidence intervals to define an acceptance region for the null hypothesis that the metrics are unbiased estimators of subjective refraction. Only two metrics (A=PFCc and X=Seidel paraxial refraction) fall outside that acceptance region, indicating that all other metrics are unbiased when tested individually. Nevertheless, a clear trend towards hyperopic bias is present that is statistically highly significant by the non-parametric sign test. This slight hyperopic bias is not due to ocular refractive chromatic aberration because the inaccuracy being reported is for paired refractions determined at the same monochromatic wavelength of light.

For comparison, the efficacy of objective refraction for large, 6mm pupils is shown in Fig. 7. Several metrics appear biased for large pupils since they produced refractions that were either more myopic (inaccuracy < 0) or more hyperopic (inaccuracy > 0) than could be accounted for by the level of precision characteristic of that specific metric. However, most metrics provided unbiased estimates of refractive error even when pupils are larger and the effects of higher-order aberrations on retinal image quality are therefore greater.

Figure 7.

Efficacy of objective refraction for 6mm pupil diameter. Plotting conventions are the same as for Fig. 6. Several metrics are biased on the hyperopic side (inaccuracy >0) and others are biased on the myopic side (inaccuracy < 0) but most are unbiased. Abscissa and ordinate values are in diopters.

A comparison of the inaccuracy of objective refraction for medium and large pupils is drawn in Fig. 8. Note that the sequence of data points in Fig. 8 is identical to the sequence in Fig. 5 but the axes have different meaning in the two figures. For any given metric, the inaccuracy of objective refraction varies between subjects. The magnitude of that variation is shown in Fig. 8 by the error bars, which show ± 1 standard error of the population mean for both pupil sizes. As noted in connection with Fig. 6, statistical confidence intervals (which are double the length of the error bars) for 3mm pupils overlap the zero inaccuracy line (vertical dashed line) for most metrics. In other words, between-subject variability is too large to draw a firm conclusion that individual metrics are hyperopically biased for 3mm pupils. Two data points lie above the positive diagonal, so for those metrics (T,R) the bias of the objective refraction became increasingly hyperopic as the pupil dilated. The other 32 points lie below the diagonal, so for most metrics the bias of the objective refraction became less hyperopic as the pupil dilated, and in many cases became myopically biased. The strong linear correlation (R=0.93) between results for the two pupil sizes indicates systematic changes of inaccuracy occur when the pupil dilates. Metrics that are the most biased in the hyperopic direction for 3mm pupils are also the most hyperopically biased for 6mm pupils. Similarly, metrics that are most biased in the myopic direction for 3mm pupils are also the most myopically biased for 6mm pupils. These changes in bias are primarily due to changes in objective refraction since subjective refractions tend to be the same for both pupil sizes (Fig. 4).

Figure 8.

Comparison of the inaccuracy of objective refraction for 3 mm and 6 mm pupils. Abscissa values are taken from Fig. 6 and ordinate values are taken from Fig. 7. Symbols indicate the population mean of inaccuracy of objective refraction computed for a given metric. Error bars indicate ± 1 standard error of the population mean. Dashed line is the orthogonal least-squares regression.

3. DISCUSSION

Numerous confounding factors have made it difficult in the past to assess the absolute accuracy of objective refraction of the eye derived from measurements of ocular wavefront aberrations (Guirao & Williams, 2003, Navarro, 2009, Thibos et al., 2004). Our results show that when these factors are brought under control by using monochromatic light, cycloplegia, pupil apodization, and unbiased psychophysical methods, many metrics of image quality lead to unbiased measurements of refractive state. Accordingly, it should be possible to use these metrics in the future as unbiased surrogates for subjective refraction at the same monochromatic (635nm) wavelength, thereby achieving accurate results without the need for empirical correction factors (Campbell et al., 2006) nor detailed models of the visual process responsible for spatial acuity (Nestares, Navarro & Antona, 2003, Watson & Ahumada, 2008). However, since typical clinical refractions employ the “maximum plus” criterion and a finite target distance, they do not determine the sphere lens that maximizes image quality for an infinitely distant target. Therefore, converting Shack-Hartmann objective refractions to typical clinical refractions will require some adjustment for these factors.

The refractive state of the eye was defined in Introduction as the target vergence required to maximize retinal image quality. Implicit in this definition is the assumption that retinal image quality is being assessed in the visually-relevant plane of the entrance apertures of the cone photoreceptors. Although this implicit assumption is appropriate for a subjective method of refraction, such as the acuity refraction method used in our study, its validity for objective refraction is not obvious. As a double-pass instrument, the Shack-Hartmann wavefront aberrometer measures the eye’s aberrations for light reflected from the fundus, but these measurements are assumed to also characterize ocular aberrations for light propagating in the forward direction relevant to vision. This fundamental assumption is often referred to as a “principle of reversibility” in optics. Invoking this principle, our definition of refractive state as measured objectively by wavefront aberrometry can be restated as the target vergence required to maximize image quality in the plane of reflection of the probe beam. This rephrasing reveals a problem, however, because the fundus is a thick tissue that reflects light from many layers within the retina, choroid, and sclera (Delori & Pflibsen, 1989, Gao et al., 2008), and the distribution of these reflections will vary with wavelength (Delori & Pflibsen, 1989). If the dominant plane of reflection lies anterior or posterior to the cone apertures, then the aberrations measured objectively will not represent the aberrations relevant to vision. In short, the aberrometer may suffer from instrumentation bias (Gao et al., 2009) analogous to the artifact of retinoscopy (Charman, 1975). The potential implication for our study is that some of the unbiased refraction predictors identified in Figs. 5 & 6 may in fact be biased metrics that appear unbiased because of cancellation of instrument bias by metric bias of the opposite sign.

Current models describe the fundus as a thick reflector, with different planes having different directionality properties (Marcos, Burns & He, 1998). Some of the reflected light is guided towards the pupil center by the photoreceptors, a phenomenon known as the optical Stiles-Crawford Effect (Burns, Wu, Delori & Elsner, 1995, Gorrand & Delori, 1995), whereas light near the margins of the pupil is dominated by non-photoreceptor reflections. Recent evidence from spectral-domain optical coherence tomography, which has the necessary axial resolution (5 microns) to discriminate between retinal layers, shows that reflection at the junction of cone inner and outer segments and at the posterior tips of cone outer segments are responsible for the light guided towards the pupil center (Gao et al., 2008). Both of these reflections will emerge from the cone entrance aperture, which is the apparent source of guided light for incoherent imaging by a wavefront aberrometer even though the source of the reflections may be deeper in the retina. In contrast, light arriving at the margins of the pupil is dominated by reflections from the pigmented retinal epithelium, which lies approximately 75 microns posterior to the cone aperture (as indicated by the outer limiting membrane) in the human fovea (Polyak, 1941). Thus the dioptric separation between these two reflecting planes will be approximately 0.2 D according to the Bennett and Rabbetts schematic eye (Bennett & Rabbetts, 1998). It is important to note, however, that the Gao et al study employed a single near IR wavelength (788 nm), and we do not have comparable data at visible wavelengths. These recent results suggest that the wavefront analyzed by the aberrometer is a combination of reflections from different depths. Further support for this hypothesis obtained with a SH aberrometer confirmed that a significant fraction (25–50%) of the light exiting the pupil center is reflected by cones but almost none of the light exiting near the pupil margin of a 5mm pupil arises from cones (Gao et al., 2009).

One implication of this duplex model of fundus reflection is that the optical path length for marginal regions of the pupil will appear abnormally long because the source of the reflection lies posterior to the photoreceptors. Accordingly, the eye will appear more myopic when measured at the pupil margin compared to the pupil center. Thus the duplex nature of fundus reflection will appear to the aberrometer as positive spherical aberration of the eye’s optical system, when in fact the phenomenon is due to differential angular selectivity of fundus reflections from different layers. Similar over-estimation of other higher order aberrations are expected in eyes that lack rotational symmetry, which may have significant implications for the field of objective wavefront aberrometry in general (Gao et al., 2009). The specific implication for our study is that by overestimating the amount of positive spherical aberration of the eye, any objective method of refraction sensitive to wavefront errors at the pupil margin will tend to overestimate the amount of myopia in an eye (or underestimate the amount of hyperopia). The magnitude of this over-estimation will depend on each metric’s sensitivity to spherical aberration and the instrument’s effective pupil size, which may account for some of the increased variance of prediction errors in Fig. 7 compared to Fig. 6. At the same time, a given metric may be intrinsically biased for predicting acuity refraction if it emphasizes features of the retinal image that are not relevant to the specific psychophysical task of letter recognition. Thus we are faced with the possibility that two sources of bias (measurement and metric) with opposite sign counteract each other, yielding apparently unbiased refractions. Alternatively, the two sources of bias with the same sign may reinforce each other, leading to spuriously high levels of bias in our experiment. We investigate these two possibilities in the following section.

3.1. The source of bias in wavefront refraction

When Zernike spherical aberration increases from zero to some positive value (i.e. C₄⁰ > 0) in an eye with no other aberrations, most metrics require a correcting lens of positive power in order to maximize image quality. The reason for this seemingly counter-intuitive behavior is related to the fact that, for Zernike positive spherical aberration, the central portion of the pupil is underpowered and the marginal parts are over-powered (Cheng, Bradley, Ravikumar & Thibos, 2010). The best strategy for optimizing image quality under these circumstances is to sacrifice the marginal areas (where wavefront error is growing rapidly as the fourth power of pupil radius) in favor of improving the focus of the central pupil area by adding a positive lens. When the eye views through a positive lens, the eye+lens system appears over-powered to metrics that give equal weight to all parts of the pupil. For example, a Zernike refraction will report myopia and prescribe a negative correcting lens when C₄⁰ > 0 even when the retinal image appears optimally focused subjectively. According to human observers, the optimum lens power is somewhat less than the power required to focus the paraxial rays, hence the best correcting lens lies somewhere between the Zernike and the Seidel prescriptions (Cheng et al., 2010, Cheng et al., 2004). The problem faced in interpreting our experiments is that the measurement artifact of aberrometry may be exaggerating the amount of ocular spherical aberration, which in turn causes Zernike prescriptions to be even more negative than needed, Seidel refractions to be even more positive than needed, and possibly make some metric-based prescriptions erroneously appear unbiased.

To disambiguate metric bias in objective refraction from wavefront measurement bias, it would help to know which metrics of image quality are unbiased when the duplex nature of reflection by the human fundus is avoided. Cheng et al (Cheng et al., 2010, Cheng et al., 2004) evaluated the accuracy of the same metrics used in our current study for predicting subjective judgment of best focus for stimuli blurred by a combination of lower and higher-order aberrations. They found that when higher-order aberrations are weak, the Zernike refraction method that minimizes RMS wavefront error accurately predicted best focus. When aberrations are strong, Zernike refractions are too myopic, Seidel (paraxial) refractions are too hyperopic, but some metrics (e.g. the fraction of pupil area for which wavefront slopes are small (PFSc), or contrast of a point image (STD), or visual Strehl ratio (VSOTF)) remain accurate predictors of optimum subjective focus. Inspection of Fig. 7 confirms that these same metrics yielded unbiased refractions in the present experiments for both medium and large pupils. This suggests that any propensity by the aberrometer to overestimate the eye’s positive spherical aberration must be relatively small, otherwise that overestimation would have made these inherently unbiased metrics appear biased in the myopic direction. Nevertheless, we cannot entirely rule out the possibility that the aberrometer exaggerates the amount of higher order aberrations, thereby exaggerating the bias shown in Fig. 7. All seven metrics found to be experimentally biased in the myopic direction for large pupils (1,2,3,8,D,J,W) are metrics that are inherently biased in the myopic direction when C₄⁰ > 0 as revealed by theoretical calculations. Similarly, the metrics that are experimentally biased in the hyperopic direction (R,T,X) are inherently biased in the hyperopic direction by the introduction of positive spherical aberration. Thus any possible measurement bias in the level of C₄⁰ would exaggerate the inherent bias of some metrics.

For 3mm pupils, only two metrics (A=PFCc and X=Seidel paraxial refraction) had statistically significant bias (Fig. 6). Nevertheless, the aggregate data (Fig. 8) suggests a tendency for objective refractions to be slightly hyperopic compared to subjective refractions for 3mm pupils. This argument is weakened by the statistical correlation between objective refractions using different metrics (Thibos et al., 2004). If this hyperopic bias is real, then its source becomes of interest. One possibility is a contribution to the wavefront by light reflected at the vitreal-retinal interface, as suggested previously to account for the hyperopic bias of autorefraction (Campbell et al., 2006, Charman, 1975) and possibly retinoscopy (Millodot & O’Leary, 1980, Mutti, Ver Hoeve, Zadnik & Murphy, 1997). Another possibility is an explanation developed recently to account for the visual impact of Zernike and Seidel forms of monochromatic aberrations (Cheng et al., 2010). When positive Zernike spherical aberration is added to a diffraction-limited optical system, the pupil center becomes hyperopically defocused and the margins become myopically defocused. For this fourth-order aberration, wavefront slope varies as the cube of pupil radius. Thus as the aberration coefficient C₄⁰ increases, slope first exceeds a threshold for functional significance at the pupil margins. These slopes can be reduced, thereby improving image quality for marginal rays, by the addition of a weak negative lens. If the power of that correcting lens is adjusted to maximize the fraction of pupil area for which wavefront slope is less than threshold, then the eye would be optimally corrected by the corresponding metric (e.g. PFSt or PFSc). However conventional Zernike analysis of this corrected eye would reveal an under-powered system (C₂⁰ < 0), thus revealing a hyperopic bias relative to an observer whose acuity is determined by a pupil fraction metric. This line of reasoning breaks down for large values of C₄⁰ because a lens of sufficient power to reduce marginal slopes to below threshold causes central slopes to exceed threshold, with a net reduction of image quality. The ideal correcting lens in this case has positive power sufficient to reduce central slopes to below threshold at the expense of large supra-threshold slopes in the pupil margins. Thus the bias of a Zernike refraction changes from hyperopic to myopic as positive spherical aberration becomes stronger, as is the case when the pupil dilates from 3mm diameter to 6mm.

In summary, a simple model of fundus reflection at a single, thin layer predicts that bias of metric-based objective refractions will vary with pupil diameter by an amount and direction that depends on the sensitivity of that metric to higher-order aberrations. Our experimental finding that bias in some metric-based objective refractions grows in magnitude when the pupil expands from 3mm to 6mm is consistent with that model. However, we cannot completely rule out the possibility that this inherent bias is magnified even more by overestimations of the eye’s positive spherical aberration by a conventional Shack-Hartmann aberrometer. If light emerging from the central pupil is dominated by reflection from the cones, whereas light emerging from the pupil margins is dominated by reflection from deeper layers (e.g. retinal epithelium or choroid), then the eye’s positive spherical aberration will be overestimated leading to even greater levels of bias seen experimentally. From a clinical perspective these bias effects are relatively small (less than 0.25 D for most metrics) and functionally important only in the most demanding circumstances.

Highlights.

• Most of the Image quality metrics may be used to determine ocular refractive state for monochromatic light accurately without the need for empirical correction factors.

• There is a bias in some metric-based objective refractions which grows in magnitude when the pupil expands from 3mm to 6mm