Journal of Vision - Accuracy and precision of objective refraction from wavefront aberrations, by Thibos, Hong, Bradley, & Applegate

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.

The purpose of a conventional, ophthalmic refraction of the eye is to determine that combination of spherical and cylindrical lenses which optimizes visual acuity for distant objects. The underlying assumption of refraction is that visual acuity is maximized when the quality of the retinal image is maximized. Furthermore, it is commonly assumed that retinal image quality is maximized when the image is optimally focused. For these reasons, the endpoint of a subjective refraction is taken as an operational definition of the term “best correction” as applied to eyes.

This paper is concerned with the problem of objectively determining the best correction of an eye from measurements of wavefront aberrations. Aberrometers measure all of the eye’s monochromatic aberrations and display the result in the form of an aberration map that describes the variation in optical path length from source to retinal image through each point in the pupil. Zernike expansion of an aberration map includes the second order aberrations of defocus and astigmatism. Thus, one obvious strategy for objective refraction is to prescribe correcting lenses based on Zernike coefficients of the second-order.

Unfortunately, the problem is not solved so easily. Several studies have shown that eliminating the second-order Zernike aberrations does not necessarily optimize the subjective impression of best-focus nor the objective measurement of visual performance (Applegate, Ballentine, Gross, Sarver, & Sarver, 2003; Applegate, Marsack, Ramos, & Sarver, 2003; Guirao & Williams, 2003; Thibos, Hong, Bradley, & Cheng, 2002). Eliminating second-order Zernike aberrations is equivalent to minimizing the root mean squared (RMS) wavefront error, but this minimization does not necessarily optimize the quality of the retinal image (King, 1968; Mahajan, 1991). Thus a search has begun for alternative metrics of optical quality that are optimized by subjective refraction when higher-order aberrations are present.

A variety of problems must be solved when converting an aberration map into a prescription for corrective lenses or refractive surgery. One of the most important is a correction for the eye’s chromatic aberration. Objective aberrometers typically use infrared light, for which the eye has relatively low refractive power compared to visible light. Optical models of longitudinal chromatic aberration (Thibos, Ye, Zhang, & Bradley, 1992) can be extrapolated to estimate the difference in optical power of the eye between the measurement wavelength and some visible wavelength, but it is unclear what wavelength should be chosen as a reference for any given eye. Furthermore, since only one wavelength can be in-focus at a time, some method is needed to factor in the relative contribution of all wavelengths, each with a different amount of defocus and a different luminance, in order to objectively refract an eye for polychromatic objects.

Another sticky problem is the lack of a universally-accepted metric of image quality that could be used to establish objectively the state of optimum-focus for an aberrated eye. One purpose of this paper is to describe a variety of such metrics based on general principles described elsewhere (Cheng, Thibos, & Bradley, 2003; Williams, Applegate, & Thibos, 2004). Assuming that consensus agreement could be achieved for a metric of choice, one still needs to deal with the fact that identifying the best correction is a multi-dimensional problem in optimization. Guirao & Williams (Guirao & Williams, 2003) have described an iterative method for finding the optimum sphere, cylinder and axis parameters that optimize a metric of image quality. Other possibilities include an objective version of the clinical technique of refraction by successive elimination. A first approximation would eliminate the bulk of defocus error by correcting the eye with a spherical lens of power M, the so-called spherical equivalent. Next, the eye’s astigmatism is corrected with a cylindrical lens, followed by a fine-tuning of the spherical lens power if necessary. This is the basis of most of the methods described below.

A different kind of problem is to incorporate into the method the refractionist’s rule “maximum plus to best visual acuity” (Borisch, 1970). According to this clinical maxim, the spherical refractive error of myopic eyes should be deliberately under-corrected. The amount of under-correction is not enough to diminish visual acuity, but it is sufficient to minimize unnecessary accommodation and to maximize the usable depth of focus (DOF) at distance and near. These twin goals are achieved by prescribing a spherical lens power that is slightly less negative (in the case of myopia) or slightly more positive (in the case of hyperopia) than the lens required to make the retina conjugate to infinity. Instead, the prescribed lens conjugates the retina with a plane at the hyperfocal distance, which is the nearest distance the retina can focus on without significantly reducing visual performance for a target located at infinity (Campbell, 1957). Consequently, the eye is left in a slightly myopic state (Figure 1B), compared to an optimum correction that would place the retina conjugate to infinity (Figure 1A). Note that the diagram in Figure 1 has been simplified by assuming that any astigmatism has already been fully corrected using the appropriate cylindrical lens.

Figure 1. Two criteria for refracting the eye. (A) An optimum refraction conjugates the retina with infinity. In this case the ideal correcting lens images infinity at the eye’s far point (•). (B) A conventional refraction conjugates the fovea with the eye’s hyperfocal point (•), which lies closer to the eye by an amount equal to half the depth-of-field (DOF). In this case the correcting lens images infinity at a point (o) slightly beyond the eye’s far point and therefore the eye remains slightly myopic.

Yet another issue is the extent to which neural factors need to be taken into account when converting an aberration map into a prescription. One such neural factor is the angular sensitivity of cone photoreceptors (Enoch & Lakshminarayanan, 1991) which is commonly modeled optically by an apodization filter in the pupil plane (Bradley & Thibos, 1995; Metcalf, 1965). Post-receptoral neural processing of the retinal image affects the processing of blurred retinal images in a manner that can be modeled as a mathematical convolution of the optical point-spread function with a neural point-spread function (Thibos & Bradley, 1995). This too may be construed as a form of apodization since the effect of the convolution will be to attenuate the remote tails of a blurred point-spread function (PSF).

Recently Guirao and Williams (Guirao & Williams, 2003) described a variety of methods for quantifying the optical quality of an eye based on (1) analysis of wavefront aberrations using pupil-plane metrics and (2) analysis of retinal image quality using image-plane metrics. They reported that all five image plane metrics they considered were more accurate than two pupil-plane metrics in predicting the optimum subjective refraction for a polychromatic target for a small population of 6 eyes. Further testing was done on a large population of 146 eyes for which aberration data for a fixed, 5.7 mm pupil were available in the literature. Unfortunately, a variety of uncontrolled conditions precluded strong conclusions from this large population (e.g. possible fluctuations of accommodation, unknown pupil size during subjective refraction, binocular refractions that likely yielded sub-optimal acuity endpoints) but nevertheless the authors found a close correlation between subjective and objective refractions computed from image-plane metrics. Although visual performance during refraction presumably depended on some combination of optical and neural factors, they found that optimizing the optical image without considering neural factors led to accurate prediction of the outcome of subjective refraction. However, no assessment of the precision of these predictions was reported.

The purpose of our study was to evaluate two general approaches to converting an aberration map into a conventional sphero-cylindrical prescription. The first approach is a surface-fitting procedure designed to find the nearest sphero-cylindrical approximation to the actual wavefront aberration map. The second approach involves a virtual through-focus experiment in which the computer adds or subtracts various amounts of spherical or cylindrical wavefronts to the aberration map until the optical quality of the eye is maximized. Preliminary accounts of this work have been presented (Thibos, Bradley, & Applegate, 2002; Thibos, Hong, & Bradley, 2001).

We define the equivalent quadratic of a wavefront aberration map as that quadratic (i.e. a sphero-cylindrical) surface which best represents the map. This idea of approximating an arbitrary surface with an equivalent quadratic is a simple extension of the common ophthalmic technique of approximating a sphero-cylindrical surface with an equivalent sphere. Two methods for determining the equivalent quadratic from an aberration map are presented next.

One common way to fit an arbitrarily aberrated wavefront with a quadratic surface is to minimize the sum of squared deviations between the two surfaces. This least-squares fitting method is the basis for Zernike expansion of wavefronts. Because the Zernike expansion employs an orthogonal set of basis functions, the least-squares solution is given by the second-order Zernike coefficients, regardless of the values of the other coefficients. These second-order Zernike coefficients can be converted to a sphero-cylindrical prescription in power vector notation using Equation 1,

Curvature is the property of wavefronts that determines how they focus. Thus, another reasonable way to fit an arbitrary wavefront with a quadratic surface is to match the curvature of the two surfaces at some reference point. A variety of reference points could be selected, but the natural choice is the pupil center. Two surfaces that are tangent at a point and have exactly the same curvature in every meridian are said to osculate. Thus, the surface we seek is the osculating quadric. Fortunately, a closed-form solution exists for the problem of deriving the power vector parameters of the osculating quadratic from the Zernike coefficients of the wavefront (Thibos et al., 2002). This solution is obtained by computing the curvature at the origin of the Zernike expansion of the Seidel formulae for defocus and astigmatism. This process effectively collects all r² terms from the various Zernike modes. We used the OSA definitions of the Zernike polynomials, each of which has unit variance over the unit circle (Thibos, Applegate, Schwiegerling, & Webb, 2000). The results given in Equation 2 are truncated at the sixth Zernike order but could be extended to higher orders if warranted.

An empirical way to determine the focus error of an eye (with accommodation paralyzed) is to move an object axially along the line-of-sight until the retinal image of that object appears subjectively to be well focused. This procedure is easily simulated mathematically by adding a spherical wavefront to the eye’s aberration map and then computing the retinal image using standard methods of Fourier optics as illustrated in Movie 1. The curvature of the added wavefront can be systematically varied to simulate a through-focus experiment that varies the optical quality of the eye+lens system over a range from good to bad. Given a suitable metric of optical quality, this computational procedure yields the optimum power M of that spherical correcting lens needed to maximize optical quality of the corrected eye. With this virtual spherical lens in place, the process can be repeated for through-astigmatism calculations to determine the optimum values of J₀ and J₄₅ needed to maximize image quality. If necessary, a second iteration could be used to fine-tune results by repeating the above process with these virtual lenses in place. However, the analysis reported below did not include a second iteration.

Movie 1. Dynamic simulation of the through-focus method of objective refraction. To determine the optimum value M of a spherical defocusing lens, a pre-determined sequence of M-values are used to modulate the wavefront map in the same way that a real lens alters the eye’s wavefront aberration function. From the new aberration map we compute the retinal point-spread function (PSF), optical transfer function (OTF), and retinal image of an eye chart. Scalar metrics of optical quality are used to optimize focus (M). The process is then repeated to optimize astigmatism parameters J₀, J₄₅. This example is for an eye with 0.1 mm of spherical aberration.

The computational method described above captures the essence of clinical refraction by mathematically simulating the effects of sphero-cylindrical lenses of various powers. Our method is somewhat simpler to implement than that described by Guirao & Williams (Guirao & Williams, 2003) who used an iterative searching method to determine that combination of spherical and cylindrical lenses which maximizes the eye’s optical quality. Regardless of which searching algorithm is used, a suitable metric of optical quality is required as a merit function. Guirao and Williams used 5 such metrics of image quality. In Appendix A we expand their list to 31 metrics by systematically pursuing three general approaches to quantifying optical quality: (1) wavefront quality, (2) retinal image quality for point objects, and (3) retinal image quality for grating objects. Implementation of image sharpness metrics for extended objects, such as a letter chart, (Fienup & Miller, 2003; Hultgren, 1990) have been left for future work. Several of the implemented metrics include a neural component that takes into account the spatial filtering of the retinal image imposed by the observer’s visual system. Strictly speaking, such metrics should be referred to as metrics of neuro-optical quality or visual quality, but for simplicity we use the term “optical quality metric” generically. For each of these 31 metrics we used the virtual refraction procedure described above to determine (to the nearest 1/8 D) the values of M, J₀ and J₄₅ required to maximize the metric. These objective refractions were then compared with conventional subjective refractions. A listing of acronyms for the various refraction methods is given in Table 1.

To judge the success of an objective method of refraction requires a gold standard for comparison. The most clinically relevant choice is a subjective refraction performed for Sloan letter charts illuminated by white light. Accordingly, we evaluated our objective refractions against the published results of the Indiana Aberration Study (Thibos et al., 2002). That study yielded a database of aberration maps for 200 eyes that were subjectively well-corrected by clinical standards. The methodology employed avoided the problems mentioned above that limited the conclusions drawn by Guirao & Williams. A brief summary of the experimental procedure used in the Indiana Aberration Study is given next.

Subjective refractions were performed to the nearest 0.25D on 200 normal, healthy eyes from 100 subjects using the standard optometric protocol of maximum plus to best visual acuity. Accommodation was paralyzed with 1 drop of 0.5% cyclopentalate during the refraction. Optical calculations were performed for the fully dilated pupil, which varied between 6-9 mm for different eyes. The refractive correction was taken to be that sphero-cylindrical lens combination which optimally corrected astigmatism and conjugated the retina with the eye’s hyperfocal point (Figure 1b). This prescribed refraction was then implemented with trial lenses and worn by the subject during subsequent aberrometry (l=633 nm). This experimental design emphasized the effects of higher-order aberrations by minimizing the presence of uncorrected second-order aberrations. The eye’s longitudinal chromatic aberration was taken into account by the different working distances used for aberrometry and subjective refraction as illustrated in Figure 2. Assuming the eye was well focused for 570 nm when viewing the polychromatic eye chart at 4 m, the eye would also have been focused at infinity for the 633nm laser light used for aberrometry (Thibos et al., 1992).

Since all eyes were corrected with spectacle lenses during aberrometry, the predicted refraction was M = J₀ = J₄₅ = 0. The level of success achieved by the 33 methods of objective refraction described above was judged on the basis of precision and accuracy at matching these predictions (Figure 3). Accuracy for the spherical component of refraction was computed as the population mean of M as determined from objective refractions. Accuracy for the astigmatic component of refraction was computed as the population mean of (Bullimore, Fusaro, & Adams, 1998) vectors. Precision is a measure of the variability in results and is defined for M as twice the standard deviation of the population values, which corresponds to the 95% limits of agreement (LOA) (Bland & Altman, 1986). The confidence region for astigmatism is an ellipse computed for the bivariate distribution of J₀ and J₄₅. This suggests a definition of precision as the geometric mean of the major and minor axes of the 95% confidence ellipse.

In our view, accuracy and precision are equally important for refraction. A method that is precise but not accurate will yield the same wrong answer every time. Conversely, a method that is accurate but not precise gives different answers every time and is correct only on average. Thus we seek a method that is both accurate and precise. However, one might argue that lack of accuracy implies a systematic bias that could be removed by a suitable correction factor applied to any individual eye. One way to obtain such a correction factor is to examine the population statistics of a large number of eyes, as we have done in this study. Any systematic bias obtained for this group could then be used as a correction factor for future refractions, assuming of course that the individual in question is well represented by the population used to determine the correction factor. Although this may be an expedient solution to the problem of objective refraction, it lacks the power of a theoretically sound account of the reasons for systematic biases in the various metrics of optical quality.

The two methods for determining the equivalent quadratic surface for a wavefront aberration map gave consistently different results. A frequency histogram of results for the least-squares method (Figure 4A) indicated an average spherical refractive error of M = –0.39 D. In other words, this objective method predicted the eyes were, on average, significantly myopic compared to subjective refraction. To the contrary, the method based on paraxial curvature matching (Figure 4B) predicted an average refractive error close to zero for our population. Both methods accurately predicted the expected astigmatic refraction as shown by the scatter plots and 95% confidence ellipses in Figure 5.

Figure 5. Scatter plots of (A) the least-squares fit of the wavefront over the entire pupil and (B) paraxial curvature matching methods of determining the two components of astigmatism. Circles show the results for individual eyes, green cross indicates the mean of the 2-dimensional distribution, and ellipses are 95% confidence intervals. Precision is the geometrical mean of the major and minor axes of the ellipse.

Computer simulation of through-focus experiments to determine that lens (either spherical or astigmatic) which optimizes image quality are computationally intensive, producing many intermediate results of interest but too voluminous to present here. One example of the type of intermediate results obtained when optimizing the pupil fraction metric PFWc (see Table 1 for a list of acronyms) is shown in Figure 6A. For each lens power over the range –1 to +1 D (in 0.125 D steps) a curve is generated relating RMS wavefront error to pupil radius. Each of these curves crosses the criterion level (l/4 in our calculations) at some radius value. That radius is interpreted as the critical radius since it is the largest radius for which the eye’s optical quality is reasonably good. The set of critical radius values can then be plotted as a function of defocus, as shown in Figure 6B. This through-focus function peaks at some value of defocus, which is taken as the optimum lens for this eye using this metric. In this way the full dataset of Figure 6 is reduced to a single number.

N	Acronym	Brief Description
1	RMSw	Standard deviation of wavefront
2	PV	Peak-valley
3	RMSs	RMSs: std(slope)
4	PFWc	Pupil fraction for wavefront (critical pupil)
5	PFWt	Pupil fraction for wavefront (tessellation)
6	PFSt	Pupil fraction for slope (tessellation)
7	PFSc	Pupil fraction for slope (critical pupil)
8	Bave	Average Blur Strength
9	PFCt	Pupil fraction for curvature (tessellation)
10	PFCc	Pupil fraction for curvature (critical pupil)
11	D50	50% width (min)
12	EW	Equivalent width (min)
13	SM	Sqrt(2nd moment) (min)
14	HWHH	Half width at half height (arcmin)
15	CW	Correlation width (min)
16	SRX	Strehl ratio in space domain
17	LIB	Light in the bucket (norm)
18	STD	Standard deviation of intensity (norm)
19	ENT	Entropy (bits)
20	NS	Neural sharpness (norm)
21	VSX	Visual Strehl in space domain
22	SFcMTF	Cutoff spat. freq. for rMTF (c/d)
23	AreaMTF	Area of visibility for rMTF (norm)
24	SFcOTF	Cutoff spat. freq. for rOTF (c/d)
25	AreaOTF	Area of visibility for rOTF (norm)
26	SROTF	Strehl ratio for OTF
27	VOTF	OTF vol/ MTF vol
28	VSOTF	Visual Strehl ratio for OTF
29	VNOTF	CSOTF vol/ CSMTF vol
30	SRMTF	Strehl ratio for MTF
31	VSMTF	Visual Strehl ratio for MTF
32	LSq	Least squares fit
33	Curve	Curvature fit

Similar calculations were then repeated for other eyes in the population to yield 200 estimates of the refractive error using this particular metric. A frequency histogram of these 200 values similar to those in Figure 4 was produced for inspection by the experimenters. Such histograms were then summarized by a mean value, which we took to be a measure of accuracy, and a standard deviation, which (when doubled) was taken as a measure of precision.

The accuracy and precision of the 31 methods for objective refraction based on optimizing metrics of optical quality, plus the two methods based on wavefront fitting, are displayed in rank order in Figure 7. Mean accuracy varied from –0.50 D to +0.25 D. The 14 most accurate methods predicted M to within 1/8 D and 24 methods were accurate to within 1/4 D. The method of paraxial curvature matching was the most accurate method, closely followed by the through-focus method for maximizing the wavefront quality metrics PFWcand PFCt. Least-squares fitting was one of the least accurate methods (mean error = -0.39 D).

Figure 7. An example of intermediate results for the through-focus calculations needed to optimize the pupil fraction metric PFWc. (A) The RMS value is computed as a function of pupil radius for a series of defocus values added to the wavefront aberration function of this eye. The pupil size at the intersection points of each curve with the criterion level of RMS are plotted as a function of lens power in (B). The optimum correcting lens for this eye is the added spherical power that maximized the critical pupil diameter (and therefore maximized PFWc) which in this example is +0.125 D.

Precision of estimates of M ranged from 0.5 to 1.0 D. A value of 0.5 D means that the error in predicting M for 95 percent of the eyes in our study fell inside the confidence range given by the mean ± 0.5 D. The most precise method was PFSc (±0.49D), which was statistically significantly better than the others (F-test for equality of variance, 5% significance level). Precision of the next 14 methods in rank ranged from ±0.58D to ±0.65D. These values were statistically indistinguishable from each other. This list of the 15 most precise methods included several examples from each of the three categories of wavefront quality, point-image quality, and grating-image quality. Rank ordering of all methods for predicting defocus is given in Table 2.

	Accuracy		Precision
Rank	Metric	Mean	Metric	2xSTD
1	PFCc	0.2406	PFSc	0.4927
2	Curv	-0.006	AreaOTF	0.5803
3	PFWc	-0.0063	VSOTF	0.5806
4	PFCt	-0.0425	PFWc	0.5839
5	SFcMTF	-0.0425	LIB	0.5951
6	LIB	-0.0681	NS	0.5961
7	VSX	-0.0731	VSMTF	0.5987
8	SFcOTF	-0.0737	EW	0.6081
9	CW	-0.0912	SRX	0.6081
10	EW	-0.1006	AreaMTF	0.6112
11	SRX	-0.1006	PFCt	0.6213
12	VSMTF	-0.1131	STD	0.63
13	NS	-0.1144	SFcMTF	0.6343
14	VOTF	-0.125	VSX	0.6391
15	PFSc	-0.1281	D50	0.6498
16	VNOTF	-0.1575	CW	0.6558
17	AreaMTF	-0.165	PFWt	0.6575
18	STD	-0.1656	PFSt	0.6577
19	VSOTF	-0.1794	RMSw	0.6702
20	SROTF	-0.1875	SFcOTF	0.6786
21	HWHH	-0.200	SRMTF	0.6888
22	PFSt	-0.2162	SROTF	0.69
23	AreaOTF	-0.2269	ENT	0.6987
24	SRMTF	-0.2544	LSq	0.7062
25	D50	-0.2825	HWHH	0.7115
26	PFWt	-0.3231	RMSs	0.7159
27	ENT	-0.3638	Curv	0.7202
28	RMSw	-0.3831	SM	0.7315
29	LSq	-0.3906	VNOTF	0.7486
30	RMSs	-0.425	Bave	0.7653
31	SM	-0.4319	PV	0.7725
32	PV	-0.4494	VOTF	0.8403
33	Bave	-0.4694	PFCc	0.9527

A similar process was used to determine the accuracy for estimating astigmatism. We found that all methods except one (PFCc) had a mean error across the population of less than 1/8 D. This accuracy is the best we could reasonably expect, given that the subjective refractions and the virtual refractions used to predict subjective refractions were both quantized at 1/8 D of cross-cylinder power. Precision of astigmatism predictions was typically better than precision for predicting defocus. The precision of all metrics for predicting astigmatism ranged from ±0.32D to ±1.0D and the 15 best methods were better than ±0.5D. Rank ordering of all methods for predicting astigmatism is given in Table 3.

	Accuracy		Precision
Rank	Metric	Mean	Metric	2xSTD
1	HWHH	0.0155	LSq	0.3235
2	LIB	0.0164	PFSc	0.3315
3	PFCt	0.0192	Bave	0.3325
4	AreaMTF	0.0258	RMSs	0.3408
5	ENT	0.0273	RMSw	0.3429
6	NS	0.0281	Curv	0.3568
7	VSX	0.03	PFWc	0.3639
8	PFSt	0.0305	PV	0.4278
9	AreaOTF	0.0313	VSMTF	0.4387
10	EW	0.0343	AreaMTF	0.4423
11	SRX	0.0343	NS	0.4544
12	SRMTF	0.038	PFCt	0.4715
13	VSMTF	0.0407	STD	0.4752
14	STD	0.0422	PFWt	0.4923
15	CW	0.0576	SM	0.4967
16	RMSs	0.0589	SRMTF	0.5069
17	VSOTF	0.0594	EW	0.5181
18	PFSc	0.0608	SRX	0.5181
19	D50	0.0665	CW	0.5287
20	SM	0.0668	LIB	0.535
21	Bave	0.0685	AreaOTF	0.5444
22	SROTF	0.0724	SFcMTF	0.5659
23	PFWc	0.0745	VSX	0.5813
24	VOTF	0.0787	VSOTF	0.6796
25	LSq	0.0899	HWHH	0.6796
26	RMSw	0.0909	SROTF	0.7485
27	Curv	0.0913	PFSt	0.7555
28	PV	0.098	SFcOTF	0.7821
29	PFWt	0.1039	VNOTF	0.816
30	VNOTF	0.1059	D50	0.8416
31	SFcOTF	0.113	ENT	0.8751
32	SFcMTF	0.1218	VOTF	0.9461
33	PFCc	0.8045	PFCc	1.0005

In comparing the precision for predicting defocus and astigmatism we found that 7 metrics were in the top-15 list for both types of prediction. Five of these were also accurate to within 1/8 D for predicting both defocus and astigmatism. Thus 5 metrics (PFSc, PFWc, VSMTF, NS, and PFCt) emerged as reasonably accurate and among the most precise. Three of these successful metrics were pupil plane metrics and two were image plane metrics. These results demonstrate that accurate predictions of subjective refractions are possible with pupil plane metrics. However, such metrics do not include the process of image formation that occurs in the eye, a process that must influence subjective image quality. For this reason, image-plane metrics of visual quality are more germane to vision models of the refraction process that seek to capture the subjective notion of a well-focused retinal image (Williams, Applegate, & Thibos, 2004).

One implication of the results presented above is that different methods of objective refraction that yield similar refractions on average are likely to be statistically correlated. We tested this prediction by computing the correlation coefficient between all possible pairs of methods for predicting M. The resulting correlation matrix is visualized in Figure 8. For example, the left-most column of tiles in the matrix represents the Pearson correlation coefficient r between the first objective refraction method in the list (RMSw) and all other methods in the order specified in Table 1. Notice that the values of M predicted by optimizing RMSw are highly correlated with the values returned by methods 3 (RMSs), 8 (Bave), 19 (ENT), and 32 (least-squares fit). As predicted, all of these metrics are grouped at the bottom of the ranking in Figure 7. To the contrary, refractions using RMSw are poorly correlated with values returned by methods 4 (PFWc), 9 (PFCt), 21 (VSX), 24 (SFcOTF), and 33 (Curvature fit). All of these metrics are grouped at the top of the ranking in Figure 7, which further supports this connection between accuracy and correlation. A similar analysis of the correlation matrix for astigmatism parameters is not as informative because there was very little difference between the various methods for predicting J₀ and J₄₅.

Another interesting feature of Figure 8 is that some refraction methods (e.g. PFCc, VOTF, VNOTF) are very poorly correlated with all other methods. This result for metric PFCc is explained by the fact that PFCc was the only metric to produce hyperopic refractions in the vicinity of M=+0.25D. However, this argument does not apply to the other two examples that are poorly correlated with most other metrics even though these other metrics produced similar refractions on average (e.g. 20 (NS), 7 (PFSc), and 23 (AreaMTF)). This result suggests that maximizing metrics VOTF and VNOTF optimizes a unique aspect of optical and visual quality that is missed by other metrics. In fact, these two metrics were specifically designed to capture infidelity of spatial phase in the retinal image.

The least-squares method for fitting an aberrated wavefront with a spherical wavefront is the basis of Zernike expansion to determine the defocus coefficient. The failure of this method to accurately predict the results of subjective refraction implies that the Zernike coefficient for defocus is an inaccurate indicator of the spherical equivalent of refractive error determined by conventional subjective refractions. On average, this metric predicted that eyes in our study were myopic by -0.39D when in fact they were well corrected.

To the contrary, matching paraxial curvature accurately predicted the results of subjective refraction. This method is closely related to the Seidel expansion of wavefronts because it isolates the purely parabolic (r²) term. It also corresponds to a paraxial analysis since the r² coefficient is zero when the paraxial rays are well focused. Although this method was one of the least accurate methods for predicting astigmatism, it nevertheless was accurate to within 1/8D. The curvature method was one of the most precise methods for predicting astigmatism but was significantly less precise than some other methods for predicting defocus. For this reason it was eliminated from the list of 5 most precise and accurate methods.

We do not know why the various metrics have different amounts of systematic bias, but at least two possibilities have already been mentioned. First, to undertake the data analysis we needed to make an assumption about which wavelength of light was well focused on the retina during subjective refraction with a polychromatic stimulus. We chose 570 nm as our reference wavelength based on theoretical and experimental evidence (Charman & Tucker, 1978; Thibos & Bradley, 1999) but the actual value is unknown. Changing this reference wavelength by just 20 nm to 550 nm would cause a 0.1 D shift in defocus, which is a significant fraction of the differences in accuracy between the various metrics.

A second source of bias may be attributed to the difference between optimal and conventional refraction methods. The objective refraction procedures described in this paper are designed to determine the optimum refraction (Figure 1a) whereas the subjective refractions were conventional (Figure 1b). The difference between the two endpoints is half the depth-of-focus (DOF) of the eye. The DOF for subjects in the Indiana Aberration Study is unknown, but we would anticipate a value of perhaps ±0.25D (Atchison, Charman, & Woods, 1997) which is about half the total range of focus values spanned in Figure 7. Accordingly, we may account for the results in Figure 7 by supposing that the curvature matching technique happens to locate the far end of the DOF interval (which is located at optical infinity in a conventional refraction) whereas some middle-ranking metric (such as VSOTF) locates the middle of the DOF, located at the hyperfocal distance. This inference is consistent with the fact that most eyes in the Indiana Aberration Study had positive spherical aberration. Such eyes have less optical power for paraxial rays than for marginal rays. Consequently, the retina will appear to be conjugate to a point that is beyond the hyperfocal point if the analysis is confined to the paraxial rays.

The preceding arguments suggest that the superior accuracy of the curvature method for determining the spherical equivalent of a conventional refraction is due to a bias in this method that favors the far end of the eye’s DOF. In short, curvature matching (and several other metrics with similar accuracy) is a biased method that successfully predicts a biased endpoint. By the same argument, the biased curvature method is not expected to predict astigmatism accurately because conventional refractions are unbiased for astigmatism. Although this line of reasoning explains why the paraxial curvature method will locate a point beyond the hyperfocal point, we lack a convincing argument for why the located point should lie specifically at infinity. Perhaps future experiments that include measurement of the DOF as well as the hyperfocal distance will clarify this issue and at the same time help identify objective methods for determining the hyperfocal distance.

Pursuing the above line of reasoning suggests that some metric near the bottom of the accuracy ranking, such as RMSw, locates the near end of the DOF. This accounting is consistent with the findings of Guirao and Williams (Guirao & Williams, 2003) and of Cheng et al. (Cheng, Bradley, & Thibos, 2004) that the optimum focus lies somewhere between the more distant paraxial focus and the nearer RMS focus. Taken together, the least-squares and curvature fitting methods would appear to locate the two ends of the DOF interval. While perhaps a mere coincidence, if this intriguing result could be substantiated theoretically then it might become a useful method to compute the DOF from the wavefront aberration map for individual eyes.