Purpose: To determine the impact of higher-order monochromatic aberrations on lower-order subjective sphero-cylindrical refractions. Methods: Computationally-aberrated, monochromatic Sloan letters were presented on a high luminance display that was viewed by an observer through a 2.5mm pupil. Through-focus visual acuity (VA) was determined in the presence of spherical aberration (*Z*_{4}^{0}) at three levels (0.10, 0.21 and 0.50D). Analogous through-astigmatism experiments measured visual acuity in the presence of secondary astigmatism (*Z*_{4}^{±2}) or coma (*Z*_{3}^{−1}). Measured visual acuity was correlated with 31 different metrics of image quality to determine which metric best predicts performance for degraded retinal images. The defocus and astigmatism levels that optimized each metric were compared with those that produced best visual acuity to determine which metric best predicts subjective refraction. Results: Spherical aberration, coma and secondary astigmatism all reduced VA and increased depth of focus. The levels of defocus and primary astigmatism that produced the best performance varied with levels of spherical aberration and secondary astigmatism, respectively. The presence of coma, however, did not affect cylindrical refraction. Image plane metrics, especially those that take into account the neural contrast sensitivity threshold (e.g. the visual Strehl ratio, VSOTF), are good predictors of visual acuity in both the through-focus and through-astigmatism experiments (*R* = −0.822 for VSOTF). Subjective sphero-cylindrical refractions were accurately predicted by some image-quality metrics (e.g., pupil fraction, VSOTF and standard deviation of PSF light distribution). Conclusion: Subjective judgment of best focus does not minimize RMS wavefront error (Zernike defocus = 0), nor create paraxial focus (Seidel defocus = 0), but makes the retina conjugate to a plane between these two. It is possible to precisely predict subjective sphero-cylindrical refraction for monochromatic light using objective metrics.

*Z*

_{2}

^{0}), which means that the RMS wavefront error was not minimized by a subjective refraction. Similar results were found by Guirao et al who reported that refraction based on minimum RMS wavefront error made the eye myopic while refraction based on paraxial focus made the eye hyperopic (Guirao & Williams, 2003). In addition, in a previous study (Cheng, Bradley, Thibos, & Ravikumar, 2003) we found that, in the presence of spherical aberration, visual acuity was better with paraxial focus than with the defocus that minimized RMS. Although subjective refractions achieve approximate paraxial focus, a systematic paraxial defocus still exists (Thibos, Hong et al., 2002). Thus, these previous studies all indicate that subjective refractions in aberrated eyes do not minimize RMS, and probably do not achieve paraxial focus, which poses the question of what optical characteristic or image quality property is optimized during a subjective refraction?

^{2}) gamma-corrected, rear-projection screen viewed through an interference filter (

*λ*= 556nm). The luminance of this monochromatic stimulus viewed by the subject was 264cd/m

^{2}. Subjects viewed the simulated images through a unit magnification relay telescope, which conjugated a 2.5mm artificial pupil centered on the primary line of sight with the subject’s entrance pupil plane. The 2.5mm artificial pupil ensured that the subject’s eye was approximately diffraction-limited (confirmed experimentally) and was computationally convenient. In this way, visual acuity could be evaluated with controlled levels of monochromatic aberrations.

*Z*

_{2}

^{0}) in the presence of three possible levels of spherical aberration (

*Z*

_{4}

^{0}) when computing blurred stimuli. Similarly, we examined the impact of astigmatic defocus (

*Z*

_{2}

^{±2}) in the presence of secondary astigmatism (

*Z*

_{4}

^{±2}) or coma (

*Z*

_{3}

^{−1}). These experiments are, therefore, controlled analogues of the standard subjective refraction in which through-focus or through-astigmatism plots of acuity are obtained.

*Z*

_{4}

^{0},

*Z*

_{4}

^{±2}, or

*Z*

_{3}

^{−1}) with 7 or 8 of levels of Zernike defocus (

*Z*

_{2}

^{0}) or astigmatism (

*Z*

_{2}

^{±2}) for a 5mm pupil diameter. The three levels of higher-order aberrations were 0.10D, 0.21D and 0.50D of equivalent defocus, where equivalent defocus (in diopters) is given by the following equation (Thibos, Hong et al., 2002).

*n*= 31) of optical quality metrics for every aberration condition used experimentally. Detailed descriptions of these metrics are given in the accompanying paper (Thibos, Hong et al., 2004). A simple clarification of the acronyms is given in Table 2 in the Appendix). These metric values were correlated with logMAR acuity to evaluate their success at predicting visual performance. Since logMAR and metrics are both dependent variables, we used principal component analysis to determine the orthogonal regression line. We also determined the level of defocus or astigmatism that optimized each metric to evaluate the accuracy and precision with which each metric predicted subjective best focus.

^{rd}order coma (0.00D and 0.50D) tested. The letter images within each row reflect the different levels of spherical defocus (panel A) or astigmatism (panels B, C and D) used to generate the through-focus plots of visual acuity. Through-focus logMAR acuities are shown within panels A, C & D for the three different levels of 4

^{th}order aberrations (triangles = 0.10D, circles = 0.21D and squares = 0.50D) and for two subjects (solid and open symbols). Only one subject and two levels (0.00 and 0.50D) of coma were tested.

^{th}order spherical aberration (

*Z*

_{4}

^{0}) impacts the visual effect of defocus (

*Z*

_{2}

^{0}) in several ways. First, the best achievable visual acuity deteriorates with increasing levels of spherical aberration. Second, confirming Applegate et al. (Applegate, Marsack, & Ramos, 2003), the amount of Zernike spherical aberration influenced the amount of defocus needed to produce the best visual acuity. Third, increasing levels of Zernike spherical aberration significantly decreased the change in logMAR produced by defocus and thus increased the depth of focus. The results in panels C&D show that the presence of secondary astigmatism (

*Z*

_{4}

^{±2}) had a similar influence on the visual effect of second order astigmatism (

*Z*

_{2}

^{±2}). The presence of coma also reduced the best achievable visual acuity and increased the depth of focus. It did not, however, change the level of astigmatism necessary to achieve best acuity. When MAR approached its minimum levels (at “best focus”), one subject’s acuity was consistently 0.1 log units better than the other. We assume this reflects a genuine neural difference between these two subjects.

*n*= 31) of optical and image quality metrics. Metric amplitude was then correlated with logMAR visual acuity observed for these same conditions (all of the data in Figure 1A, C and D). The scattergrams in Figure 2 show 3 examples of optical and image-quality metrics, which were reasonably well correlated with logMAR visual acuity (left panels) and 3 examples that were poorly correlated (right panels). In each case (good correlation and poor correlation), we illustrate examples of wavefront-based, PSF-based and OTF-based metrics. The metric of “pupil fraction” (PFSt, the fraction of pupil area for which the optical quality of the eye is reasonably good, Cheng, Thibos, & Bradley, 2003) was well correlated with acuity (

*R*= −0.837), whereas RMS wavefront error (RMSw) was poorly correlated (

*R*= 0.493). Also, two PSF metrics produced different correlations: the standard deviation of intensity values in the PSF, normalized to diffraction-limited value (STD,

*R*= −0.816) and PSF half-width-at-half height (HWHH,

*R*= 0.365). Finally, visual Strehl ratio (VSOTF, the contrast-sensitivity-weighted OTF divided by contrast-sensitivity-weighted OTF for diffraction limited optics, Cheng, Himebaugh, Kollbaum, Thibos, & Bradley, 2004) correlated well with logMAR (

*R*= −0.822), whereas the metric designed to capture the phase changes in the image, OTF/MTF ratio (VOTF), was poorly correlated with acuity (

*R*= −0.182). It is interesting to see that the metric of VOTF successfully divided the data points in to two distinct parts. The data points at the bottom area of Figure 2F represent visual acuity obtained under the aberration conditions that introduced large phase shifts (VOTF < 1), whereas the data points at the top area of Figure 2F represent visual acuity obtained under aberration conditions that maintained phase (VOTF ≈ 1). For those aberrations that did not introduce phase shifts (VOTF ≈ 1), visual acuity deteriorated due to decreased contrast, which had little effect on VOTF. Therefore it is not surprising to see many of the data points spread horizontally around the VOTF value of 1. Interestingly, we found that good visual acuity was also obtained from some aberration conditions with large phase shifts. Good acuities under these conditions are possible because these large phase shifts resulted in multiple ghost images (e.g. sample images in Figure 1C and D) each of which could be resolved. Such fortuitous legibility is less likely to occur with multiple letter presentation and overlapping ghost images. Future studies designed specifically to examine the importance of spatial phase might clarify the utility of the metric VOTF for predicting the impact of phase shifts on vision.

*Z*

_{4}

^{0}. Two examples show that the RMS wavefront error (RMSw) and the PSF half-width-at-half-height (HWHH) are clearly optimized by a defocus level that does not provide maximum acuity. However, visual Strehl ratio (VSOTF) was optimized by almost the exact level of defocus that produced maximum visual acuity. Therefore, if RMSw or HWHH were used as a basis for objective refractions, less than optimal visual acuity would ensue whereas if VSOTF was used, optimal visual acuity would be achieved.

*Z*

_{4}

^{0}and the meridionally varying

*Z*

_{4}

^{±2}. We confirmed that the level of

*Z*

_{4}

^{0}and

*Z*

_{4}

^{±2}had a profound impact on the subjective spherical and cylindrical refraction (Figure 1), and that some but not all objective metrics of optical quality predicted subjective best focus with great accuracy over a wide range of aberration levels (Figure 4 and Table 1).

*Z*

_{4}

^{0}) and it did not minimize RMS wavefront error. In our simulated aberration paradigm, we also found that subjective refractions (level of positive

*Z*

_{2}

^{0}required for maximum acuity) were affected by the levels of

*Z*

_{4}

^{0}, and maximum acuity was not achieved by minimal RMS. In both studies, acuity was maximized in the presence of positive spherical aberration by a positive

*Z*

_{2}

^{0}, which was also reported in two studies by Applegate et al. (Applegate, Marsack, & Ramos, 2003; Applegate, Sarver, & Khemsara, 2002). Thus, relative to the defocus that minimizes RMS, both real and virtual eyes were myopic when acuity was maximized. In both cases, this reflects a shift in subjective refraction toward the spherical power required to focus paraxial rays.

*Z*

_{2}

^{0}) and for paraxial focus (zero

*r*

^{2}) (Figure 1A). This difference may reflect the influence of pupil apodization (Stiles-Crawford effect (Zhang, Ye, Bradley, & Thibos, 1999)), which biases visual responses toward the pupil center (Charman, Jennings, & Whitefoot, 1978; Koomen, Scolnik, & Tousey, 1951; Koomen, Tousey, & Scolnik, 1949; Thibos, Hong et al., 2002) in the real eye subjective refractions. In addition, as argued by Thibos et al (Thibos, Hong, et al., 2004), clinical subjective refractions are designed to bring the hyperfocal distance rather than infinity into focus (Figure 5A), therefore coincidentally rendering paraxial rays from an infinite target well focused. This coincidence indicates that half of the depth of focus (dioptric difference between hyperfocal distance and infinity) is approximately equal to the dioptric difference between optimal focus and paraxial focus. However, in the current study, we aimed to maximize visual acuity at a target distance of infinity, therefore the retina is conjugated to a point at infinity, and is conjugated to a point beyond infinity (hyperopia) for paraxial rays (Figure 5B).

Metric # | Metric acronym | R | SE error (D) | Astigmatic error (D) |
---|---|---|---|---|

1 | WF(1): RMSw | 0.4931 | 0.5383 | 0.4642 |

2 | WF(2): PV | 0.3759 | 0.5383 | 0.5392 |

3 | WF(3): RMSs | 0.5375 | 0.6083 | 0.5375 |

4 | WF(4): PFWc | −0.6999 | 0.1583 | 0.1092 |

5 | WF(5): PFWt | −0.7293 | 0.505 | 0.0942 |

6 | WF(6): PFSt | −0.8374 | 0.0417 | 0.1025 |

7 | WF(7): PFSc | −0.8016 | 0.095 | 0.1392 |

8 | WF(8): Bave | 0.5783 | 0.7283 | 0.6542 |

9 | WF(9): PFCt | −0.7434 | 0.295 | 0.3242 |

10 | WF(10): PFCc | −0.3308 | 0.4617 | 0.4625 |

11 | PS(1): D50 | 0.5452 | 0.4583 | 0.3558 |

12 | PS(2): EW | 0.7377 | 0.375 | 0.1508 |

13 | PS(3): SM | 0.536 | 0.655 | 0.4942 |

14 | PS(4): HWHH | 0.3652 | 0.6617 | 0.1292 |

15 | PS(5): CW | 0.4879 | 0.455 | 0.3042 |

16 | PS(6): SRX | −0.7053 | 0.3917 | 0.1442 |

17 | PS(7): LIB | −0.7301 | 0.1383 | 0.2092 |

18 | PS(8): STD | −0.8158 | 0.0483 | 0.1408 |

19 | PS(9): ENT | 0.7198 | 0.6217 | 0.3358 |

20 | PS(10): NS | −0.8464 | 0.0283 | 0.1242 |

21 | PS(11): VSX | −0.7942 | 0.075 | 0.1625 |

22 | SF(1): SFcMTF | −0.5338 | 0.1683 | 0.4175 |

23 | SF(2): AreaMTF | −0.7445 | 0.035 | 0.1508 |

24 | SF(3): SFcOTF | −0.6416 | 0.1583 | 0.2308 |

25 | SF(4): AreaOTF | −0.7671 | 0.0317 | 0.1442 |

26 | SF(5): SROTF | −0.6542 | 0.3917 | 0.1608 |

27 | SF(6): VOTF | −0.1815 | 0.385 | 0.1975 |

28 | SF(7): VSOTF | −0.8216 | 0.0717 | 0.1508 |

29 | SF(8): VNOTF | −0.2938 | 0.2283 | 0.4992 |

30 | SF(9): SRMTF | −0.7658 | 0.3883 | 0.2625 |

31 | SF(10): VSMTF | −0.8456 | 0.075 | 0.1542 |

32 | Paraxial | 0.525 | 0.5925 |

RMSw | root-mean-squared wavefront error computed over the whole pupil (micrns) |
---|---|

WF(2): PV = | peak-to-valley difference (microns) |

WF(3): RMSs = | root-mean-squared wavefront slope computed over the whole pupil (arcmin) |

WF(4): PFWc = | pupil fraction when critical pupil is defined as the concentric area for which RMSw < criterion (e.g. wavelength/4) |

WF(5): PFWt = | pupil fraction when a “good sub-aperture satisfies the criterion PV < criterion (e.g. wavelength/4) |

WF(6): PFSt = | pupil fraction when a "good sub-aperture satisfies the criterion horizontal slop and vertical slop are both < criterion (e.g. 1 arcmin) |

WF(7): PFSc = | pupil fraction when critical pupil is defined as the concentric area for which RMSs < criterion (e.g. 1 arcmin) |

WF(8): Bave = | average blur strength (diopters) |

WF(9): PFCt = | pupil fraction when a “good’ sub-aperture satisfies the criterion Bave < criterion (e.g. 0.25D) |

WF(10): PFCc = | pupil fraction when critical pupil is defined as the concentric area for which Bave < criterion (e.g. 0.25D) |

PS(1): D50 = | diameter of a circular area centered on peak which captures 50% of the light energy (arcmin) |

PS(2): EW = | equivalent width of centered PSF (arcmin) |

PS(3): SM = | square root of second moment of light distribution (arcmin) |

PS(4): HWHH = | half width at half height (arcmin) |

PS(5): CW = | correlation width of light distribution (arcmin) |

PS(6): SRX = | Strehl ratio computed in spatial domain |

PS(7): LIB = | light-in-the-bucket |

PS(8): STD = | standard deviation of intensity values in the PSF, normalized to diffraction-limited value |

PS(9): ENT = | entropy of the PSF |

PS(10): NS = | Neural sharpness |

PS(11): VSX = | visual Strehl ratio computed in the spatial domain |

SF(1): SFcMTF = | spatial frequency cutoff of radially-averaged modulation-transfer function (rMTF) |

SF(2): AreaMTF = | area of visibility for rMTF (normalized to diffraction-limited case) |

SF(3): SFcOTF = | spatial frequency cutoff of radially-averaged optical-transfer function (rOTF) |

SF(4): AreaOTF = | area of visibility for rOTF (normalized to diffraction-limited case) |

SF(5): SROTF = | Strehl ratio computed in frequency domain (OTF method) |

SF(6): VOTF = | volume under OTF normalized by the volume under MTF |

SF(7): VSOTF = | visual Strehl ratio computed in frequency domain |

SF(8): VNOTF = | volume under neurally-weighted OTF, normalized by the volume under neurally weighted MTF |

SF(9): SRMTF = | Strehl ratio computed in frequency domain (MTF method) |

SF(10): VSMTF = | visual Strehl ratio computed in frequency domain (MTF method) |