Metrics of optical quality derived from wave aberrations predict visual performance

Abstract

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

Introduction

Visual performance is defined by how well a visual task of interest can be performed by a given individual or group of individuals. A classic but not the only test of visual performance is high-contrast visual acuity. 

Recently, the link between visual performance and optical quality of the eye has enjoyed a renewed interest, due largely to the development of clinically viable wavefront aberrometers and the popularization of wavefront guided refractive surgery. Currently, the most common method for describing the wavefront error of the eye is the normalized Zernike expansion (Thibos, Applegate, & Schweigerling, 2000). The Zernike expansion is in common use for several reasons. First, it provides an efficient way to specify an entire wavefront aberration map with a relatively small set of Zernike coefficients. Second, individual Zernike basis functions (i.e., modes) correspond to classical optical aberrations, such as defocus, astigmatism, coma, and spherical aberration. Third, when normalized by the recommended OSA system, the Zernike functions are mutually orthogonal, and the root mean squared (RMS) wavefront error of each function is given by its coefficient. Consequently, a Zernike expansion provides a convenient accounting scheme in which the total RMS wavefront error is equal to the square root of the sum of the squares of the individual coefficients in the Zernike spectrum of a wavefront aberration map. 

Over a large range of RMS errors (an equivalent dioptric range of around 3 diopters), visual acuity decreases with increasing RMS error of the corneal first surface (Applegate et al., 2000). However, at low levels of whole eye aberrations (less than 0.25 equivalent D), the RMS wavefront error cannot account for an observed two-line variation in visual performance (Applegate, Marsack, Ramos, & Sarver, 2003). Closing this gap in our understanding of the visual consequences of low levels of residual wave aberration is important to fully realize the potential of custom refractive surgery as well as customized contact lens corrections. 

The complex interactions of wave aberrations at low levels of optical error and how these interactions impact visual performance are being systematically investigated by our laboratory (Applegate, Ballentine, Gross, Sarver, & Sarver, 2003; Applegate, Sarver, & Khemsara, 2002; Applegate, Marsack et al., 2003) and others (Cheng, Bradley, Thibos, 2004). We typically perform these experiments using the Zernike expansion to describe wave aberration and keep the amount of wave aberration purposely low (RMS < = 0.25 µm over a 6-mm pupil — a dioptric equivalent of equal or less than 0.19 D) (G. Pettit, personal communication). We first explored the visual impact of low levels of aberration by observing how a fixed amount of RMS error loaded into single Zernike modes (2^nd through 4^th radial orders) impact letter acuity of an individual (Applegate et al., 2002). In these experiments, each subject served as his or her own control. That is, we measured how a change in aberration altered visual performance as measured by high-contrast logMAR acuity. These experiments revealed that 0.25 µm of aberration over a 6-mm pupil reduced visual acuity by an amount that depended on which Zernike mode contained the wavefront error. Modes near the center of each radial order have a greater impact on visual performance (more letters lost) than modes near the edge of the pyramid. This result is seen in Figure 1. 

Figure 1

 

The number of high-contrast letters lost as a result of loading 0.25 μm of RMS error into each Zernike mode in the 2^nd through 4^th radial order individually. Notice that Zernike modes in the center of each order affect vision more than the modes near the edge of each order. (Figure 1 adapted from data presented in Applegate et al., 2002.)

In a second experiment, 6 levels of RMS wavefront error (0.00 µm, 0.05 µm, 0.10 µm, 0.15 µm, 0.20 µm, and 0.25 µm) were loaded one at a time into the same 12 Zernike modes to determine how they affected high-contrast logMAR acuity (Applegate, Ballentine et al., 2003). The results showed that within any given Zernike mode, performance decreased linearly with increasing RMS wavefront error and reconfirmed the results of the first experiment by demonstrating that Zernike modes near the center of each radial order impacted acuity more (increased slopes of the linear fit) than modes near the edge of each radial order. 

However, real eyes do not exhibit single-mode aberrations (Howland & Howland, 1977; Porter, Guirao, Cox, & Williams, 2001; Thibos, Hong, Bradley, & Cheng, 2002). To individually study all or even most of the possible or even relevant combinations of aberrations and magnitudes present in a normal population would be impractical. Instead, we chose to systematically approach the problem one step at a time by exploring interactions of various combinations of two aberration modes. Accordingly, a third experiment was conducted to investigate how low levels of RMS wavefront error split between two Zernike modes affect visual acuity (Applegate, Marsack et al., 2003). The experiment was performed by varying the relative proportion of the wavefront error attributable to each of two Zernike modes while keeping total RMS wavefront error constant at 0.25 micrometers over a 6-mm pupil used in the previous experiments. Figure 2 illustrates the relative proportions of each Zernike mode used in the combination. 

Figure 2

 

Relative contribution of Zernike coefficient values for two paired modes in the nine test conditions. Total RMS error for the combined modes was held constant at 0.25 µm. Four sets of coefficients were paired in each of the nine individual test conditions for a total of 34 unique test conditions (4 pairs times 9 conditions with = 0.25 µm repeated). The pairings studied were (defocus and spherical aberration), (astigmatism and secondary astigmatism), (spherical aberration and quadrafoil), and (spherical aberration and secondary astigmatism). (Figure 2 constructed from data presented in Applegate, Marsack et al., 2003.)

The experimental result revealed a variation in high-contrast visual acuity of nearly two lines on a log MAR chart, despite the fact that the total RMS error was held constant at 0.25 micrometers over a 6-mm pupil (a fixed equivalent dioptric error of 0.19 D). The magnitude of the loss was dependent on which aberration modes were combined and in what ratio. This finding demonstrates that the manner in which the Zernike modes are combined significantly impacts measured acuity in a way that RMS wavefront error and equivalent dioptric error cannot predict. The likely reason is that RMS wavefront error specifies only the standard deviation of the wavefront error over the pupil. It does not contain any information as to how this wavefront error was distributed within the pupil, the resulting effect on the point spread function in the spatial domain, or the impact on the modulation transfer function or the phase transfer function in the frequency domain. 

That equivalent diopters and RMS error cannot account for any of the two-line changes in acuity induced in this study and that these combinations are challenging due to interactions between Zernike modes that affect visual perception make this data set an interesting step in developing single-value metrics predictive of visual performance. 

The study reported here uses this latter data set (Applegate, Marsack et al., 2003) to investigate the ability of 31 scalar metrics derived from wave aberration maps to predict changes in high-contrast logMAR acuity. Phrased as a question, we ask, “Can the change in visual acuity induced by different combinations of Zernike modes be well predicted by one or more of the 31 metrics derived from the wavefront aberration map?” 

Methods

The 31 metrics used in this study are mathematical functions that have as input the normalized Zernike expansion coefficients and as output a single value. As seen in “Appendix A” of the accompanying article in this issue (Thibos, Hong, Bradley, & Applegate, 2004), the 31 metrics tested can be classified into two types: pupil plane metrics and image plane metrics. Pupil plane metrics are defined by qualities of the shape of the wave aberrations in the pupil plane. The image plane metrics can be subdivided as metrics based on the point spread function or metrics based on the optical transfer function. 

Ten of the 31 metrics considered are pupil plane metrics (PPM). Additionally, 11 image plane metrics are based on the point spread function (PSFM) and 10 image plane metrics are based on the optical transfer function (OTFM). The Zernike data remain the fundamental wave aberration input in all cases. In some metrics, neural weighting has been added to mimic effects of the neural system, providing a fuller description of the visual process. For a full mathematical description of each metric, the reader is referred to “Appendix A” of the accompanying article in this issue by Thibos et al. (2004). 

The metric values are derived from the wavefront maps, and, therefore, are dependent on the Zernike spectra used in each experimental condition; they are not subject to experimental variance because they were fixed and contained in the aberrated chart. Thus it is appropriate to treat the metric values as independent variables for correlation analysis. Accordingly, conventional linear regression was performed on each of 31 scatter plots to determine the ability of each metric to predict changes in visual acuity. 

Results

The correlation coefficients for all 31 metrics are displayed in Figure 3. 

Figure 3

 

R-squared values for all metrics tested. “Appendix A” of the accompanying article appearing in this feature (Thibos et al., 2004) contains the names and description of each metric. The visual Strehl calculated using the optical transfer method provides the highest correlation and can account for 81% of the variance in acuity.

In the sections that follow, we will present the experimental results for the best and worst metric for each class of metrics acting on our dataset. 

Pupil plane metrics (PPMs)

Of the PPMs, metric PFSt (pupil fraction satisfying the requirement that the magnitude of the local slope is less than a fixed criterion [1 arcmin]) was the best predictor of visual performance (Figure 4, R² = 0.69), whereas RMSw (wavefront error) was least predictive. Because RMS values were fixed at 0.25 µm in the Applegate, Marsack et al. (2003) experiment, RMSw has zero predictive power for variation in visual acuity as is evident in Figure 5. 

Figure 4

 

The best PPM tested is PFS_t: pupil fraction satisfying the requirement that the magnitude of the local slope is less than a fixed criterion (1 arcmin). Plotting letters lost as a function of pupil fraction reveals an R² = 0.69.

Figure 5

 

Letters lost (letters read on the aberrated chart — letters read on the unaberrated chart) for a fixed level of RMS error. Varying the mix of a variety of pairs of Zernike terms, while keeping total RMS wavefront error constant at 0.25 µm over a 6-mm pupil, varies acuity over nearly 2 lines (10 letters). RMS has zero ability to predict visual performance in this scenario where RMS error is constant in all test cases.

Point spread function metrics (PSFM)

The PSFM that best correlates with visual performance is VSX, the visual Strehl computed in the spatial domain (Figure 6, R² = 0.76), whereas the least correlated PSFM is HWHH, the half width at half height of the PSF (Figure 7, R² = 0.01). 

Figure 6

 

The best PSFM tested is the visual Strehl computed in the spatial domain (VSX). Plotting letters lost as a function of visual Strehl reveals an R² = 0.76.

Figure 7

 

The worst PSFM tested is HWHH: the PSF half width at half height. Plotting letters lost as a function PSF half width at half height reveals an R² = 0.01

Optical transfer function metrics (OTFM)

The best OTFM is VSOTF: the visual Strehl computed in the frequency domain (Figure 8, R² = 0.81), whereas the worst is VOTF: the volume under the OTF/volume under the MTF (Figure 9, R² = 0.18). 

Figure 8

 

The best OTFM tested is VSOTF: the visual Strehl computed in the spatial frequency domain. Plotting letters lost as a function of visual Strehl reveals an R² = 0.81

Figure 9

 

The worst OTFM tested is VOTF: volume under the OTF normalized by the volume under the MTF. Plotting letters lost as a function of VOTF reveals an R² = 0.18.

Although only the best and worst metrics were chosen for graphical presentation, other metrics computed in this study did well at predicting visual performance and are displayed in Figure 3. There are 6 metrics that accounted for 70% or more of the variance in logMAR acuity. 

The results reported above are calculated for the average visual performance of three observers. What is clinically more relevant to the patient and clinician is the ability of a metric to predict the change in visual performance induced by a change in aberration structure of an individual. For example, a gain/loss of acuity might be induced by a change in wave aberration due to refractive surgery or a custom contact lens. To examine individual correlations, the metrics were run on the individual data of each of the three subjects. Figure 10 shows the correlation coefficients for the average and individual subjects on the most predictive metric from each group of metrics. While the correlation is higher in the average case, it is still excellent for all individual cases. Considering the dioptric equivalent for all test conditions was held constant at 0.19 D (RMS = 0.25 µm), the results demonstrate that the better metrics are very good at predicting how a change in wave aberration affects high-contrast logMAR acuity. 

Figure 10

 

Coefficients of correlation for the individual subjects and the average coefficient of correlation for all subjects for the best metric in each category: best pupil plane metric PPM, best point spread function metric PSFM, and best optical transfer function metric OTFM.

Discussion

Because all of the metrics are designed to measure some aspect of optical quality of the eye, we anticipated that the metrics would be correlated with each other, as well as being correlated with visual performance. In particular, the three metrics called visual Strehl ratio (VSOTF, VSMTF, and VSX as seen in Figure 3), all capture the effectiveness of the retinal PSF at stimulating the neural portion of the visual system relative to the effectiveness of a perfect (i.e., diffraction-limited) PSF. However, each of these 3 metrics is computed slightly differently rendering different answers. The method for computing VSX weights the optical PSF with a neural weighting function that is centered on the PSF peak and then determines the peak of the neural-weighted PSF. To the contrary, VSOTF is computed in the spatial frequency domain by weighting the Fourier transform of the PSF (i.e., the OTF) with the neural contrast sensitivity function. The spatial counterpart to this operation would locate the neural PSF at the origin, not the peak of the PSF, and then compare the value of the neural-weighted PSF at the origin with the diffraction-limited case. Some of the PSFs that correspond to the Zernike spectra used in the Applegate, Marsack et al. (2003) experiments were decentered from the origin established by the pupil; therefore, it is not surprising that they had different power for predicting changes in acuity. 

An additional caution is warranted. The wave aberrations used in the experiments of Applegate, Marsack et al. (2003), while typical of the magnitude of aberration present in real eyes, did not have a distribution of Zernike modes that is typical in normal or abnormal eyes. Specifically, it would be unlikely for a real eye to exhibit error in only two modes because real eyes tend to have some error across many modes (Howland & Howland, 1977; Porter et al., 2001; Thibos et al., 2002). 

Wave aberration is only one factor in retinal image quality (other factors include scatter and chromatic aberration), and retinal image quality is only one factor contributing to visual performance. Therefore, we anticipate that metrics of image quality will not perform as well in accounting for differences in visual performance across individuals as they do in predicting the impact of a change in wave aberration within an individual. This is not a liability of this experimental design; instead, it is a strength. The experimental design is intended to predict how an induced change in aberration affects the visual performance of an individual. This is exactly what an individual patient and clinician want to know: how a change in aberration structure (e.g., induced by refractive surgery) affects an individual patient. 

Comparison with other studies

Although some metrics performed very well at predicting high-contrast acuity while others performed very poorly, the results might be quite different for other visual tasks, such as contrast sensitivity or face recognition. One of the challenges for the future is to determine whether any metric, or combination of metrics, can adequately predict visual performance for a variety of visual tasks under a variety of stimulus conditions. Preliminarily, this can be achieved by comparing three studies in this issue. 

Thibos et al. (2004), Cheng et al. (2004), and this work are published studies in this special issue of JOV that predict visual performance using the same set of metrics under different viewing conditions. It is encouraging to learn that VSOTF, one of the better metrics in our study of polychromatic visual performance, also performed well for predicting monochromatic performance in the Cheng et al. (2004) study as well as in predicting sphero-cylindrical refraction in the Thibos et al. (2004) study. These three studies also agree on which metrics do not predict acuity well (RMS_w, HWHH, and VOTF). 

Conclusions

A variety of single-valued metrics can be derived using wavefront error measurements of the eye. Here we report on 31 of these metrics. Six of the 31 single-value metrics based on wavefront error accounted for over 70% of the variance in high-contrast log MAR acuity. The best single — alue metric reported in this study for predicting how a change in aberration affects high-contrast logMAR visual acuity was the visual Strehl calculated using the OTF method. The visual Strehl accounted for 81% of the average variance in high-contrast logMAR visual acuity. 

Acknowledgments

This work was supported in part by NIH Grant EY R01 05280 (RAA), NIH grant EY R01 05109 (LNT), training fellowship T32 EY07024 (JDM and RAA), Core Grant NIH/NEI EY07551 to the College of Optometry, University of Houston, University of Houston HEAF funds, and The Visual Optics Institute at the College of Optometry, University of Houston. 

Commercial relationships: LNT and RAA have submitted a provisional patent on metrics of wavefront aberration. 

Corresponding author: Jason D. Marsack. 

Email: . 

Address: College of Optometry, 4800 Calhoun, University of Houston, Houston, TX 77204. 

Appendix

•  
  RMSw — root-mean-squared wavefront error computed over the whole pupil (µm).
•  
  PV — peak-to-valley difference (µm).
•  
  RMSs — root-mean-squared wavefront slope computed over the whole pupil (arcmin).
•  
  Bave — average blur strength (diopters)
•  
  PFWc — pupil fraction when critical pupil is defined as the concentric area for which RMSw < criterion (λ/4).
•  
  PFSc — pupil fraction when critical pupil is defined as the concentric area for which RMSs < criterion (1 arcmin).
•  
  PFCc — pupil fraction when critical pupil is defined as the concentric area for which Bave < criterion (0.25 D).
•  
  PFWt — pupil fraction when a “good” sub-aperture satisfies the criterion PV < criterion (λ/4).
•  
  PFSt — pupil fraction when a “good” sub-aperture satisfies the criterion horizontal slope and vertical slope are both < criterion (1 arcmin).
•  
  PFCt — pupil fraction when a “good” sub aperture satisfies the criterion Bave < criterion (0.25 D).
•  
  D50 — diameter of a circular area centered on PSF peak, which captures 50% of the light energy (arcmin).
•  
  EW — equivalent width of centered PSF (arcmin).
•  
  SM — square root of second moment of light distribution (arcmin).
•  
  HWHH — half width at half height (arcmin).
•  
  CW — correlation width of light distribution (arcmin).
•  
  SRX — Strehl ratio computed in spatial domain.
•  
  LIB — light in the bucket.
•  
  STD — standard deviation of intensity values in the PSF, normalized to diffraction-limited value.
•  
  ENT — entropy of the PSF inspired by an information —theory approach to optics (Guirao & Williams, 2003).
•  
  NS — neural sharpness (Williams, 2003).
•  
  VSX — visual strehl ratio computed in the spatial domain.
•  
  SFcMTF — spatial frequency cutoff of radially averaged modulation-transfer function.
•  
  SFcOTF — cutoff spatial frequency of radially averaged optical transfer function.
•  
  AreaMTF — area of visibility for rMTF, normalized to diffraction-limited case.
•  
  AreaOTF — area of visibility for rOTF, normalized to diffraction-limited case.
•  
  SRMTF — Strehl ratio computed in frequency domain, MTF method.
•  
  SROTF — Strehl ratio computed in frequency domain, OTF method.
•  
  VSMTF — visual Strehl ratio computed in frequency domain, MTF method.
•  
  VSOTF — visual Strehl ratio computed in frequency domain, OTF method.
•  
  VOTF — volume under OTF normalized by the volume under MTF.
•  
  VNOTF — volume under neurally weighted OTF, normalized by the volume under neurally weighted MTF.