# Visual Impact of Zernike and Seidel Forms of Monochromatic Aberrations

Purpose. The aim of this study was to examine the impact of different aberrations modes (e.g., coma, astigmatism, spherical aberration [SA]) and different aberration basis functions (Zernike or Seidel) on visual acuity (VA).

Methods. Computational optics was used to generate retinal images degraded by either the Zernike or Seidel forms of second through fourth-order aberrations for an eye with a 5-mm pupil diameter. High contrast, photopic VA was measured using method of constant stimuli for letters displayed on a computer-controlled, linearized, quasimonochromatic (λ = 556 nm) display.

Results. Minimum angle of resolution (MAR) varied linearly with the magnitude (root mean square error) of all modes of aberration. The impact of individual Zernike lower- and higher-order aberrations (HOAs) varied significantly with mode, e.g., arc minutes of MAR per micrometer of root mean square slopes varied from 7 (spherical defocus) to 0.5 (quadrafoil). Seidel forms of these aberrations always had a smaller visual impact. Notably, Seidel SA had 1/17th the impact of Zernike SA with the same wavefront variance, and about 1/4th the impact of Zernike SA with matching levels of r^{4} wavefront error. With lower-order components removed, HOAs near the center of the Zernike pyramid do not have a large visual impact.

Conclusions. The majority of the visual impact of high levels of fourth-order Zernike aberrations can be attributed to the second-order terms within these polynomials. Therefore, the impact of SA can be minimized by balancing it with a defocus term that flattens the central wavefront (paraxial focus) or maximizes the area of the pupil with a flat wavefront. Over this wide range of aberration types and levels, image quality metrics based on the Point Spread Function (PSF) and Optical Transfer Function (OTF) can predict VA as reliably as VA measures can predict retests of VA, and, thus, such metrics may become valuable predictors of both VA and, via optimization, refractions.

*MD, PhD

^{†}PhD

^{‡}PhD, FAAO

School of Optometry, Indiana University, Bloomington, Indiana (XC, AB, SR, LNT), and Vistakon Corporation, Jacksonville, Florida (XC).

Received October 6, 2009; accepted December 21, 2009.

In the clinical environment, visual acuity (VA) testing is the gold standard, albeit indirect method for assessing optical quality of the retinal image. This surrogate test of retinal image quality forms the basis of subjective refractions for the simple reason that the VA is relatively easy to measure, and it varies in direct proportion to the level of defocus, modulated only by pupil size (MAR = blur/4, where blur = blur circle diameter = defocus in diopters (D), modulated only by pupil size (P), (MAR (arc min) = blur (arc min)/4, and blur circle diameter (radians) = defocus (diopters) × pupil diameter (meters).^{1–4} In addition to the well-documented impact of lower-order aberrations (LOAs) on VA, more recent studies have shown that higher-order aberrations (HOAs) can also degrade VA.^{5–8} Accordingly, VA continues to be used clinically as a surrogate measure of image quality when assessing the impact of HOAs (e.g., best spectacle-corrected VA in studies of postrefractive surgery, intraocular implants, and keratoconic eyes).^{9–11}

Increased levels of HOAs introduced by multifocal (aberrated) intraocular lenses,^{12} multifocal contact lenses,^{13} aberrated custom contact lenses,^{14} corneal disease,^{15} and refractive surgery^{16–18} can all lead to reduced VA. Conversely, correcting HOAs using deformable mirrors,^{7,19} custom-lathed optical corrections,^{20} and aspheric intraocular lenses^{11} can improve VA. It appears, however, that not all HOAs have the same ability to degrade vision. Studies using computationally blurred letter charts^{5} or deformable mirrors^{8,21,22} have shown that individual Zernike modes closer to the center of the Zernike pyramid (lower meridional frequencies) have more impact than those near the edge (higher meridional frequencies).^{23}

Although individual Zernike polynomials (modes) are considered as individual aberrations, they contain multiple terms. For example, the “spherical aberration” (SA) Zernike polynomial includes both r^{4} and r^{2} terms and defines the following wavefront error (WFE)

Consequently, the WFE in the central 50% of the pupil is dominated not by the r^{4} term (which is about zero in the central region of the pupil), but by the opposite sign r^{2} term (Fig. 1). This structure of individual Zernike modes becomes important because experimental studies have shown that, in eyes with SA, best VA (subjective) refractions for circular pupils are dominated by the central optics.^{24–26} These results suggest that VA is determined primarily by the WFE in the central portion of the pupil, and optimum VA is achieved by flattening the central wavefront. It is clearly possible, therefore, that the observed effect of Zernike SA on VA is due not to the r^{4} component of the polynomial (the component that makes it SA and is primarily responsible for the WFE at the pupil margins), but rather to the r^{2} term generating spherical defocus-like wavefronts in the pupil center. The Seidel convention for defining monochromatic aberrations defines SA as a^{0}_{4}r^{4}, and thus, lacks the r^{2} term that dominates the central region of Zernike SA (Fig. 1).

The present study was designed to reexamine the relationship among different types of HOAs and VA, specifically investigating the role of the basis functions for defining aberrations (e.g., Zernike vs. Seidel). We examined the hypothesis that the visual impact of individual Zernike HOAs stems primarily from the lower-order components included in each mode. Although it is possible that the complexity of the monochromatic HOAs and their interaction may preclude any simple relationship between VA and aberration level, we also look for pupil plane and image plane metrics that allow accurate prediction of VA in the presence of both LOAs and HOAs.

## METHODS

### Stimulus Generation

We studied the visual impact of monochromatic HOAs on vision using the approach first described in detail by Burton and Haig.^{27} This method uses computationally blurred stimuli, which generate retinal images in the experimental subject's eye that are degraded only by diffraction and the specific monochromatic HOAs included in the computational model.^{28,29} To use this method accurately, the optical degradation of the display monitor and the subject's eye must be precompensated within the blur calculations by deconvolving the computed image by the display Modulation Transfer Function (MTF) and the observer's eye OTF. Partial implementation of the Burton and Haig method has been used previously to examine the impact of HOAs on VA.^{5,30}

All optical computations used a 5-mm pupil. The display non-linearity and its two-dimensional MTF as well as the subject's OTF were measured and precompensated in the image calculations. Monochromatic aberrometry measurements (Wavefront Sciences, COAS) across the central 2.5-mm pupil confirmed that the subjects' OTFs were within a couple of percent of diffraction limit, and thus, during the experiment, subjects used a 2.5-mm artificial pupil, and their optics was represented in the precompensation process by the diffraction-limited OTF of a 2.5-mm pupil. The display MTF was determined empirically by displaying a series of high-contrast, sinusoidal gratings and then measuring the displayed contrast and phase using SpectraScan PR714 (Photo Research Inc.) radiometer configured with a two-pixel wide, rectangular measurement window oriented parallel to the gratings. These measurements were corrected for the OTF of the measurement window and extrapolated to the cutoff frequency of the display. Separate MTFs for horizontal and vertical gratings were radially interpolated to produce a two-dimensional MTF used to precompensate the stimulus.

Accuracy of image computation was verified using a number of benchmarks. Corrections for display non-linearity (gamma correction) and low-pass filtering were confirmed experimentally. The images generated from wavefronts defined in terms of Zernike basis vectors (our methods) were identical to those generated using van Meeteren's methods.^{31} MTFs and PSFs for diffraction-limited cases were closely predicted by diffraction theory. Spherical defocus (1 diopter) generated image contrast minima and phase reversals at the same spatial frequencies as expected from geometrical optics theory. Also, by imaging the computer display through a simple model eye, computationally blurred images were compared directly with optically blurred images. Images were captured by a charge-coupled device (the model retina), with the optical system well focused and the displayed images blurred or with well-focused displayed images and a defocused optical system. As shown in Fig. 2, the results from the two methods were essentially identical for both spherical and astigmatic blur, thus confirming the accuracy of our computational blurring methods.

### Psychophysical Experiment

High luminance (264 cd/m^{2}) quasimonochromatic (17 nm full width at half height) test stimuli were generated with a digital projector and rear projection screen viewed through an interference filter (peak transmission, 556 nm). Three adult subjects participated in this study. Each experiment was performed on two subjects, and all data reported are the means of the two subjects. All subjects had corrected VA better than 20/20. They were carefully refracted in the instrument and viewed the display through a unit magnification telescope, which imaged the 2.5-mm artificial pupil into the geometric center of the subject's natural pupil, which was stabilized using a bite-bar (Fig. 3). To ensure that computed image and neural image were matched in size, a unit magnification telescope was set up with a 2.5-mm artificial aperture that was located 2.5 m from the 512 × 512 pixel display, which subtended 3.26°. This geometry ensured that the spatial frequencies within the OTF were the same as those in the stimulus amplitude spectrum. The artificial pupil of the system was conjugated and centered with the entrance pupil of the subject's eye.

The computational compensation for the display MTF and eye OTF essentially acts as high-pass filtering of the computed image (Fig. 4). Such filtering prevented us from using high-contrast stimuli and high spatial frequencies (need to divide by very small numbers near to display or eye cutoff), and thus, all VA measurements were band-limited to 38 cpd (20/16 VA) and generated with 30 to 21% contrasts to ensure that the precompensated images remained within the eight-bit dynamic range of the system. Intensity profiles and amplitude spectra of a 30% contrast 20/40 letter (thin solid line) imaged with 0.05 μm of SA (Fig. 4A, B) emphasize the increase in the intensity range of the displayed letter when it has been precompensated by the deconvolution process described earlier (dashed lines). The figure also shows the difference between the desired retinal image (thick solid line) and the one that would be achieved if the deconvolution process had not been implemented (dotted line).

For comparison, retinal images were also computed for the same stimuli if no precompensation was performed, and the stimuli were viewed polychromatically. In the latter case, the retinal image is blurred by the display and the eye's optics and differs substantially from the desired image. Control experiments show that, if the precompensation is not implemented, VA can be degraded by slightly more than 0.1 log units by the additional degradation of display and subject's OTF. The impact of the additional filtering varies with the level of filtering generated by the computational optics.

We used a forced-choice method of constant stimuli (alternative forced choice [4-AFC] and 10-AFC for the tumbling E and Sloan letter acuity, respectively) to generate psychometric functions of percent correct identification vs. letter size. A random sequence of 8 to 12 letter sizes (0.05 logMAR steps) with 10 or 20 presentations per letter size constituted a single experimental run that generated a single psychometric function. A fixation box (30% contrast twice the size of the largest letter presented within a given trial) was presented throughout the trial to aid fixation. All letters were dark on a light background with a stroke width of 1/5th the letter height and a 1:1 aspect ratio. Individual aberrated letters were presented for 0.5 s (signaled by a tone) and the subject's task was to indicate which letter was present by pressing a key. To avoid learning effects, no feedback was given about correctness of responses. VA was determined by fitting the psychometric functions with a Weibull function of the form

where P is the probability of a correct answer, γ is the rate of guessing (0.25 for 4-AFC and 0.1 for 10-AFC), α is the inflection point, β is proportional to the steepness of the function when plotted on semilog coordinates, and w is the stroke width of the letter. MAR is the interpolated stroke width corresponding to criterion performance of P_{c} = 62.5% and P_{c} = 55% correct identification for the tumbling E and Sloan letters, respectively. Solving Eq. 2 for w gives MAR explicitly as

The 10 letters (C, D, H, K, N, O, R, S, V, and Z) originally used by Sloan et al.^{32} exhibit similar resolution (range about 0.15 logMAR) when intermixed in a resolution task when focused^{33,34} but are slightly more variable when defocused.^{34} Subjects provided informed consent, and the experimental protocols were approved by Indiana University institutional review board.

### Aberration Parameters

Monochromatic aberrations are often described using Seidel or Zernike formats,^{35} and these two types of aberration basis function differ in a systematic way. For example, Zernike astigmatism describes the wavefront that will create a “circle of least confusion” image (half way between either line foci) that is myopic along one primary meridian and hyperopic along the orthogonal meridian (Fig. 5). Seidel astigmatism, however, defines the wavefront that will generate one of the line foci and concentrate the defocus at the orthogonal meridian. Likewise, Zernike SA describes a wavefront that modulates its defocus sign as a function of radial distance, e.g., from a hyperopic center to a myopic surround for positive Z^{0}_{4}. Seidel SA, on the other hand, describes a wavefront that is well focused centrally and either myopic or hyperopic at the pupil margins. Not surprisingly, Seidel and Zernike aberrations generate quite different PSFs (Fig. 5). Similarly, Zernike secondary astigmatism distributes the WFE across meridian and radial position, whereas our “Seidel” form of secondary astigmatism retains one meridian and the pupil center well focused, and, thus, concentrates the WFE toward the pupil margins and along meridians orthogonal to that which is focused.

The wavefront aberration function (WAF) for Seidel SA is

where a^{0}_{4} is the Seidel coefficient of the polynomial S^{0}_{4} =r ^{4}. When balanced by defocus to minimize root mean squared WFE, the result is Zernike SA.

Thus, we can express Zernike SA using Seidel aberration coefficients as

The key feature of the two WAFs defined by Eqs. 4 and 5b is that they have identical amounts of r^{4} aberration. Thus, if the visual effects of SA are dominated by the r^{4} term, then these two WAFs should have identical visual effects. Because the r^{4} term is the only term that qualifies these WAFs as HOAs, the above prediction is the natural choice for a null hypothesis. A counter-hypothesis arises, however, if the visual effects of these WAFs are dominated by the r^{2} term. If this is true, then the WAF in Eq. 4 should have less visual impact compared with that in Eq. 5b because the r^{2} component of Zernike SA causes paraxial defocus. A third hypothesis is that image quality in eyes is determined by the amount of root mean squared WFE. If this is true, then the WAF in Eq. 4 should have a greater visual impact than that in Eq. 5b because it has fourfold larger root mean square (RMS) because of the Z^{0}_{2} required to cancel the r^{2} term within Z^{0}_{4}.

In short, by comparing the visual impact of WAFs Eqs. 4 and 5b we can test three alternative hypotheses that predict the effect of Seidel SA is smaller than, equal to, or greater than the effect of Zernike SA. A similar line of reasoning can be applied also to other modes of HOAs (e.g., fourth-order astigmatism), but not coma, for which monochromatic images generated with Zernike and Seidel coma differ only in prism (position).

A Zernike expansion of the wavefront aberration W as a function of the polar pupil coordinates (r and θ) is

where c^{m}_{n} are scalar coefficients applied to the Zernike circle polynomials Z^{m}_{n} of radial order n and meridional frequency m.^{36} A Seidel expansion is a power series^{35} written in analogous form as

where a^{m}_{n} are scalar coefficients applied to the Seidel polynomials S^{m}_{n} of radial order n and meridional frequency m. According to Eq. 6, a given Seidel aberration can be expanded into a weighted sum of Zernike aberrations, and according to Eq. 7, a given Zernike aberration can be expanded into a weighted sum of Seidel aberrations. Thus, an individual aberration in one system is a combination of aberrations in the other system.^{37}

The relationship between Seidel and Zernike aberration coefficients was established by performing a Zernike expansion of individual Seidel modes.^{35} For example, the WAF for Seidel astigmatism is

which has the Zernike expansion

Because these two expansions describe the same wavefront, the coefficients for the r^{2}cos 2θ term in Eqs. 8 and 9 must be equal. Likewise, the coefficients for the r^{2} terms must be equal. This equivalence leads to the desired relationship between the Seidel and Zernike coefficients,

By this method, we determined the combination of Zernike coefficients needed to generate all the desired Seidel aberrations.

The overall blurring strength of a WAF was quantified by root mean squared WFE, computed as the square root of the sum of squared Zernike coefficients. Micrometers of root mean squared WFE were converted to diopters of equivalent defocus M_{e}^{26} with the linear equation

Spherical blur levels in our experiments ranged from 0.23 to 3.61 μm of RMS error, which corresponds to 0.25 to 4.00 D of equivalent defocus. Examples of visual stimuli blurred by 0.5 D of Zernike and Seidel aberrations are shown in Fig. 5. The differences in the images created from these RMS-matched Zernike and Seidel aberrations are striking.

## RESULTS

### Complexities in the Psychometric Functions

A typical psychometric function (Fig. 6A) exhibits a monotonic, but non-linear relationship between stimulus strength and visual performance. When fit with a suitable function (Weibull, in our case) they are generally described by two parameters: The horizontal position of the function (α) determines threshold, whereas the slope (β) provides evidence of the experimental noise (internal to the observer or external). The Weibull function provides an excellent fit for these data, with r^{2} values in excess of 0.9. Consistent with previous studies of VA psychometric functions,^{33,38–40} we find that most psychometric functions extend over about 0.2 and 0.4 logMAR.

Because of the oscillating nature of the defocused MTF and its corresponding oscillating impact on the human contrast sensitivity function,^{41} it is possible to obtain non-monotonic psychometric functions that oscillate over a range of letter sizes (and thus spatial frequencies). Although these were rarely observed when using the Sloan 10-letter set, they were commonly encountered when measuring tumbling E VAs (Fig. 6), presumably because the E is the most grating-like of all letters. In the non-monotonic psychometric functions (Fig. 6B, C), the performance drops as image contrast dips (see inserts) and further decreases in letter size produce increased performance when the image contrast increases irrespective of whether the characteristic frequency (e.g., 2.5 c/letter) has the correct phase (Fig. 6C) or is phase reversed (Fig. 6B).

Interpretation of the non-monotonic psychometric functions is complex. We have the option of defining MAR as the smallest letter size for which performance achieves the criterion (e.g., 55% correct), or the largest letter size that drops to this criterion. We adopted the rule of recording VA as the largest letter size for which performance drops to our criterion threshold level (half way between chance perfect performance, or 55% for Sloan VA and 62.5% for tumbling E VA).

### Relationship Between Magnitude of HOAs and VA

A careful review of previous literature^{4} revealed a simple linear relationship between spherical blur magnitude and the minimum resolvable spatial detail (MAR) in which MAR = blur circle diameter/4. This simple relationship can be generalized to the spherocylindrical blur.^{3} We have examined this relationship for a series of individual Zernike aberrations (Fig. 7) and find that, for each Zernike mode, a linear relationship exists between the magnitude of the aberration (RMS) and MAR. The r^{2} values from linear regression are, with one exception, all above 0.9, with a mean value of 0.95. Linear regression of the RMS values against logMAR produced significantly lower r^{2} (p < 0.01), emphasizing that the linear relationship between blur and MAR for LOAs also holds true for the HOAs.

The results for tumbling E and Sloan VA are very similar. Although the overall impact of aberrations was generally higher for tumbling E than for Sloan, the relative impact of each mode was almost the same. As shown previously, there are striking differences between the impact of individual aberrations. We have quantified the visual impact of individual Zernike modes by the slope of the functions shown in Fig. 7 in arc minutes of MAR per micrometer of RMS, which range from 7.52 (SA and tumbling E) to 0.5 (quadrafoil, tumbling E, and Sloan). The Sloan VA slope data are plotted as a function of radial order and meridional frequency in Fig. 8 (top panel). The figure shows the impact of individual Zernike modes arranged in a Zernike pyramid. Although the highest meridional frequencies clearly have the smallest impact, there is no consistent relationship between slope and frequency (f) in that slopes for f = 0 and f = 2 are both higher than those for f = 1 (coma). The top panel also shows that there is no systematic relationship between the radial order of individual Zernike modes and their impact on VA.

Smith^{42} has shown that MAR = B/4 for spherical defocus, and B = blur circle diameter in radians = P × defocus (D). Thus, for our experiment in which a 5-mm pupil is used, the prediction is that MAR = 5 × 10^{−3}/4 radians/D or 4.3 arc minutes per diopter of defocus. Converting our Sloan VA data obtained with spherical blur from minutes per micrometer RMS to minutes per diopter using Eq. 11, we observe that spherical defocus degrades VA by 4.6 min/D. For a given RMS value, the wavefront vergence along the primary meridians of the astigmatism will be square root of 2 less than that observed with spherical defocus with the same RMS,^{43} and, thus, the associated blur size at the circle of least confusion will be square root of 2 smaller. As predicted, therefore, our data show that the impact on Sloan VA of second-order Zernike astigmatism is root 2 lower than that observed with spherical defocus (Fig. 7, top right panel) when evaluated in terms of RMS, confirming that the impact of both second-order Zernike defocus and astigmatism exhibit the same relationship in which MAR = B/4. Using the conversion from RMS to equivalent diopters (the diopters of spherical defocus that would generate a given RMS, Eq. 11), we found this approximate relationship to be generalized to Zernike SA, which had slopes of 4.6 arc minutes/D_{eq}.

### Impact of Aberration Basis Functions on Their Visual Effect

We have examined the visual impact of Zernike and Seidel versions of primary astigmatism and three aberrations of the fourth order (SA, and both forms (horizontal/vertical [H/V] and oblique) of secondary astigmatism). Sloan VAs were similarly affected by primary oblique astigmatism, with a circle of least confusion image (Zernike) or a line focus (Seidel). This was not true, however, for H/V astigmatism. Tumbling E VA was much more resistant to Seidel H/V astigmatism than to Zernike H/V astigmatism. Slopes (arc minutes of MAR per micrometer of RMS) were 16 times higher for Zernike H/V astigmatism. There was a small difference for Sloan VA too (slopes 1.6 times higher for Zernike H/V astigmatism than Seidel).

The most striking differences between Zernike and Seidel forms of individual aberrations were observed with fourth-order aberrations for both tumbling E and Sloan VA (Fig. 9). The impact of Zernike fourth-order aberrations is much greater than that of fourth-order Seidel. The Zernike slopes (arc minutes of MAR per micrometer of RMS) are 35, 17, and 29 times greater than those generated by fourth-order Seidel aberrations of oblique secondary astigmatism, SA, and H/V secondary astigmatism, respectively (Fig. 9, left column). These data emphasize the importance of basis function when defining individual aberrations, and challenge the notion that SA is a visually significant aberration. For example, the visual impact of Seidel SA (slope of the data shown in Fig. 9) is only 60% of that of the slope of Zernike quadrafoil (Fig. 7), the latter of which is generally considered to be of minor visual importance.

The data plotted in Fig. 9 allow us to test the initial hypotheses outlined in the Methods. As pointed out in the Methods, Seidel SA will have four times the RMS of the Zernike SA with matching levels of r^{4} because the coefficient of Z^{0}_{2} required to cancel the r^{2} term in Z^{0}_{4} must be square root of 15 larger than the C^{0}_{4}. Therefore, because the slopes of MAR vs. RMS differ by a factor of 17 for Zernike SA and Seidel SA, the MAR vs. r^{4} slopes will differ by 17/4 (Fig. 9, right column). That is, for matched levels of r^{4}, high levels of Zernike SA has 4.25 times the impact of Seidel SA, confirming that the visual impact of SA cannot be attributed solely to the level of r^{4} terms (reject hypothesis 1 for SA). A similar rule applies to secondary astigmatism. Our version of Seidel secondary astigmatism uses Z^{2}_{2} to cancel the r^{2}cos 2θ terms in Z^{2}_{4} to generate paraxial focus, and also adds Z^{0}_{4} and Z^{0}_{2} to generate a line focus. Because of these additional terms, the RMS of Seidel secondary astigmatism is 5.5 times greater than Zernike secondary astigmatism, with matching levels of r^{4}cos 2θ terms. Thus, in terms of r^{4}cos 2θ, a high level of Zernike secondary astigmatism is between 35/5.5 and 29/5.5 (about 5 or 6) times more visually detrimental than Seidel secondary astigmatism (Fig. 9, right column). It is clear, therefore, that although Zernike and Seidel SA and secondary astigmatism share common fourth-order terms (r^{4} for SA and r^{4}cos 2θ for secondary astigmatism), most of the visual impact of the Zernike forms of these aberrations cannot be attributed to these fourth-order terms.

The above analysis suggests that the primary factor determining the visual impact of Zernike fourth-order aberrations must be the second-order terms included within the polynomials (e.g., see Eq. 1). This hypothesis, outlined in the Methods section, also predicts that Zernike fourth-order aberrations will have a much larger visual impact than the Seidel forms, as shown to be true in Fig. 9. Although the above analysis supports the hypothesis that it is the second-order components of the fourth-order Zernike aberrations that are responsible for their large visual impact, it does not distinguish between two alternative explanations. First, the r^{2} terms within the fourth-order Zernike polynomials may be entirely responsible for their visual impact, and essentially act independent of the coexisting r^{4} terms. Second, although the r^{2} terms are primarily responsible for the visual impact of the fourth-order Zernike aberrations, their effect is not independent of the r^{4} terms. We have discriminated between these two possibilities by comparing the impact on MAR of the r^{2} WFE when alone (Z^{0}_{2}), and in the presence of r^{4} (Z^{0}_{4}). For a given level of RMS, there is square root of 15 greater r^{2} in Z^{0}_{4} than in Z^{0}_{2}. Therefore, although the Zernike Z^{0}_{2} and Z^{0}_{4} data in Fig. 7 indicate similar impact of defocus and Zernike SA on MAR (slopes of 6.1 and 5.7 min per micrometer of RMS, respectively), there is about four times more r^{2} in the Z^{0}_{4} than in the Z^{0}_{2}. This means that the r^{2} within the Z^{0}_{4} wavefront is only about 23% as visually detrimental as the r^{2} within the Z^{0}_{2}. Therefore, although the majority of the visual impact of Z^{0}_{4} comes from its second-order component, this r^{2} term has considerably less impact when in combination with the opposite sign r^{4} term.

We have plotted the visual impact of individual Seidel aberrations (arc minutes of MAR per micrometer of RMS) as a function of the meridional frequency and radial order in the bottom pyramid of Fig. 8. Note that the Seidel SA, although at the pyramid center, generates one of the smallest visual impacts. The Seidel aberration data do show that LOAs have a much larger impact than do HOAs, irrespective of their meridional frequencies. This result is consistent with the idea that the large visual impact of low meridional frequency Zernike modes is due to the lower components of these Zernike modes.

## DISCUSSION

One of the driving forces behind the investigation of HOA in human eyes is the recent technologic advances that enable contact lenses,^{44} interocular lenses,^{45} and photoablative refractive surgeries^{46} to include corrections for higher-order monochromatic aberrations. However, significant technical challenges for successful correction of ocular HOAs remain. For example, many of the HOAs are very small (e.g., ref. ^{26}) and would require submicrometer accuracy in the correction method. Also, HOA corrections can only be successful if the correction is prevented from decentering and rotating.^{47–49} These two challenges to successful correction of HOAs emphasize the need to identify those aberrations that are visually most significant and least susceptible to rotation and decentration.

Our data (Fig. 7) show that the visual impact of all the individual aberration modes increases linearly with aberration magnitude (RMS), emphasizing that the visual impact of all HOAs, like that of LOAs,^{3,4} is directly proportional to their amplitude, which in turn emphasizes the value of correcting the higher-magnitude HOAs. The fact that ocular aberration levels drop logarithmically with increasing radial order and meridional frequency^{26} suggests, therefore, that HOA corrections should concentrate on the larger third- and fourth-order aberrations of coma, trefoil, and SA. The complicating factor is that not all aberration modes have equal visual impact (Figs. 7 and 9). Even more challenging is the realization that even for a single aberration type (e.g., SA), its visual impact depends critically on the optical basis functions used to define the aberration (e.g., Seidel vs. Zernike). For example, one of the most important findings of this study is that the visual impact of SA, considered to be perhaps the most visually important HOA,^{5,22} depends critically on the basis function used to define it.

The data in Fig. 9 emphasize that the visual impact of Zernike SA is due mostly to the lower-order r^{2} term within the Zernike SA and as such is not primarily a HOA effect. The Seidel SA impact (slope [arc minutes/micrometer of RMS] = 0.3 [E's] and 0.38 [Sloan]) is only 4% to 10% that of the Zernike SA (slopes = 7.52 [E's] and 3.9 [Sloan]). Therefore, although there are two advantages of selecting SA for correction (the population mean is not zero^{26} and it will be tolerant to lens rotation errors), our Seidel results suggest that, if optimally balanced by defocus, SA may have only minor visual importance.

The SA results imply, therefore, that in the presence of high levels of SA, spherical power adjusted to ensure paraxial focus will generate improved VA compared with the same eyes with spherical power adjusted to achieve minimum RMS. That implication is consistent with results from a population of adults,^{26} indicating that a typical dilated subjective refraction (sphere power that maximizes VA) provides an approximate paraxial correction.^{50} That is, in the presence of significant SA, subjective quality for small high contrast letters is optimized by a sphere correction that approximately focuses the light passing through the pupil center. Such a prescription would, therefore, dramatically reduce the impact of ocular SA.

Previous work^{30,51} using stimulus blur methods similar to those used in the present study also found that the visual impact of SA mostly disappeared when defocus was adjusted to approximate paraxial focus. However, earlier results from our lab^{52} with lower and more typically experienced levels of SA (0.09 to 0.45 micrometer of RMS) showed that optimal focus transitioned from minimum RMS (Z^{0}_{2} = zero) with the lowest level of SA to a defocus level about half way between minimum RMS and paraxial focus with higher levels of SA. That is, in every case studied, except for the 0.09 μm of SA tested by Cheng et al.,^{52} adding some Z^{0}_{2} spherical defocus of the same sign as the Z^{0}_{4} produced significant reductions in the visual impact of the SA.

The impact of adjusting the second-order spherical defocus term (Z^{0}_{2}) on the resultant wavefront produced by SA (Z^{0}_{4}) can be seen in Fig. 10 in which we show the WFE across a 5-mm diameter pupil for three different levels of Z^{0}_{4} (0.09, 0.21, and 0.54 μm) when Z^{0}_{2} is set to zero (panel A), when Z^{0}_{2} is adjusted to obtain paraxial defocus (r^{2} terms set to zero, panel B), and when Z^{0}_{2} is adjusted to maximize the area of the pupil with a flat wavefront (WFE < ± λ/4, panel C). When SA is small (Z^{0}_{4} = 0.09 μm), setting Z^{0}_{2} to zero (the refraction that will minimize RMS) will achieve the largest pupil area with a flat wavefront. However, as SA levels rise, paraxial focus produces a larger central flat region of the wavefront, and the central flat region can be further increased by selecting a defocus term that is between the minimum RMS and paraxial refractions. This trend is quantified in Fig. 11A, which shows that the Z^{0}_{2} term required to maximize the proportion of the pupil with a flat wavefront increases as the level of SA increases, but is always slightly less than that required to achieve paraxial focus.

The impact on image quality of the three different strategies for refracting an eye with SA described in Figs. 10 and 11A was evaluated by plotting the MTF (Fig. 11B) for each condition in the presence of a high level of SA (0.54 μm). These MTFs show that with a minimum RMS refraction (set Z^{0}_{2} to zero), Z^{0}_{4} produces a defocus-like MTF which drops to zero at about 8 cpd and oscillates with accompanying phase reversals beyond this MTF zero.^{53} This MTF shares much in common with that produced by spherical defocus with the same RMS (thin lines in Fig. 11B). Unlike the defocus MTF, however, the Z^{0}_{4} MTF does not introduce the same phase reversals. In contrast, when the r^{2} term is zeroed (paraxial focus), the MTF oscillations are eliminated, emphasizing that they are produced by the r^{2} term imbedded in the Zernike SA polynomial. The defocus level between minimum RMS and paraxial focus that maximizes the flat region of the wavefront (Fig. 10C) also avoids the MTF oscillations and produces higher levels of image modulation than either the paraxial or minimum RMS refraction. Thus, for eyes with high levels of SA (e.g., eyes after refractive surgery^{17,54,55}), a refraction strategy that flattens the central wavefront should be considered. Fig. 12 shows an example image of a letter chart generated with a 6-mm pupil and 0.4 μm of Z^{0}_{4} with a minimum RMS refraction as specified by Eq. 5 (image outside of circle) compared directly with that generated with the same amount of r^{4} as specified by Eq. 4 with paraxial focus (inside of circle). This image clearly shows that Seidel SA produces a high bandwidth but low contrast image (inside circle), but Zernike SA removes the high frequencies.

The above analysis and the experimental data reported in this study emphasize that the impact of SA on VA can be minimized by careful selection of the accompanying defocus term. Indeed, a spherical refraction that maximizes the fraction of the pupil area for which the wavefront is reasonably flat will produce a full bandwidth image with high spatial frequency image contrast similar to those produced by 0.1 diopters of defocus. Thus, with an appropriate choice of spherical refraction, even large amounts of ocular SA will have little impact on high-contrast VA because of the long tail of the MTF. However, SA will lower image contrast over a wide range of spatial frequencies (Fig. 11B), and, thus, will lower contrast sensitivity over the same SF range. It is likely, therefore, that correcting SA will have a minor effect on high-contrast VA, but a more significant effect on contrast sensitivity or low-contrast VA, as has been found in recent studies.^{6,56–59}

In the present study, Zernike third-order coma, trefoil, secondary astigmatism, and quadrafoil all have a larger visual impact than Seidel SA. Because quadrafoil and secondary astigmatism tend to be very small in normal eyes,^{26} our results suggest that, as long as defocus terms can be optimized, then coma and trefoil may be the visually most important aberrations to correct, or to avoid introducing via refractive surgery.^{54,60,61}

The above analysis shows that no single wavefront characteristic (r^{4}, r^{2}, or RMS) can account for the visual impact of different modes of HOAs and LOAs or different types of the same aberration (Zernike or Seidel). These findings confirm earlier attempts that failed to predict VA as a function of HOA RMS and other characteristics of the wavefront.^{5,21,52,62} However, for both defocus and astigmatism, the single image plane metric of blur size (blur vector length)^{3} is highly correlated with the resulting VA. Also, several studies of HOAs^{21,52,62,63} have reported that there are characteristics of the image blur (PSF and OTF) that, irrespective of the aberration type, correlate highly with experimental measures of VA or perceived visual quality. Indeed, when combining image plane measures of optical quality with a formal letter recognition model, Watson and Ahumada^{64} were able to predict the actual VA observed in the presence of a range of aberration levels and types with a high degree of accuracy.

The wide range of types, magnitudes, and combinations of aberration used in this study offers an opportunity to identify image plane metrics that correlate with VA that will be generalizable to most optical situations. The current study used both Zernike and Seidel forms of aberrations originating from varying amounts of nine different modes (defocus, primary astigmatism, coma, trefoil, SA, secondary astigmatism, and quadrafoil) from three subjects, with a total of 296 VA measurements. We used a series of 31 metrics described by Thibos et al.,^{50} most of which are image plane metrics and some include neural filtering properties of the eye. Two scatter plots showing image plane optical quality metrics that offer a high predictive value for VA are shown in Fig. 13. These data include the variance contributed by two different VA tasks (tumbling E and Sloan), three observers, and the inherent variability in VA measurements. Principal component analysis produced correlation coefficients of 0.87 and 0.86, indicating that VA can be reliably predicted by these two image plane metrics. Interestingly, both metrics use neural weighting of the optical PSF (neural sharpness [NS]^{50}) and the optical MTF (visual Strehl ratio derived form the MTF [VSMTF]), and, thus, as anticipated, these data emphasize that VA is best predicted by considering both optical and neural factors. Thus, the more general linear equation (more general than Smith's LOA equation: MAR = B/4) relating image quality to MAR can be restated in terms of the neurally weighted PSF height (logMAR = 0.54 × −logNS − 0.1572) or the neurally weighted MTF (logMAR = −0.5829 × logVSMTF − 0.24). These results are similar to those observed on a previous data set.^{52}

An image quality metric that is highly correlated with VA can prove extremely valuable in that it can be used as a predictor of VA, and, thus, can act as a surrogate for actual VA testing. To assess the value of these metrics for predicting VA, we quantified the mean residual error (RMSE) expected from the linear models defined earlier. In both of the above cases, RMSE is 0.10 logMAR. That is, the mean VA error anticipated by using these linear models will be 0.1 logMAR. Interestingly, when the retinal image is degraded by defocus, test-retest differences in VA are, on average, about 0.1 logMAR.^{65} Therefore, although the above image quality metrics are not perfect at predicting VA, they are about as good at predicting VA in the presence of image blur as a standard clinical VA measure, and as such may qualify as an effective substitute for actual VA measurements, and, thus, may be used as an objective substitute for subjective refractions.^{52} Using the Cheng et al.^{52} aberration and VA data, Watson and Ahumada^{64} were able to use a template-matching model that included estimates of neural noise and customized model parameters to predict actual Sloan VA with RMSE as low as 0.056 logMAR. It is important to emphasize, however, that the image quality metrics and the VA measures used in this analysis are both monochromatic, and it has to be determined yet whether monochromatic measures of image quality can predict VA under the more typically encountered polychromatic lighting conditions.

## ACKNOWLEDGMENTS

This study was funded in part by the NIH grant NEI R01 EY 05109 awarded to Larry Thibos.

Psychophysics software was written by Kevin Haggerty. Thibos and Bradley are named co-inventors on a patent application “System and Method for Optimizing Clinical Optic Prescriptions” #10/582470 owned in part by the Indiana University.

Arthur Bradley

School of Optometry

Indiana University

800 East Atwater Avenue

Bloomington, Indiana 47405

e-mail: bradley@indiana.edu

## REFERENCES

**Keywords:**

monochromatic aberrations; visual acuity; Zernike; Seidel; retinal image quality