These results support the idea that the eye's wave aberrations are interdependent in ways that improve the eye's MTF. This positive interaction is present for the total aberrations of real eyes, but it does not occur for corneal aberrations alone and it does not occur for “synthetic” model eyes, which have aberrations drawn randomly from the experimental population. The relative advantage for the MTF of the true aberrations is prominent for larger pupils, but disappears for small pupils. Though we do not expect that all randomization schemes would produce the same results, our randomizations were chosen to maintain not just overall RMS error, but also the eye's general pattern of decreasing aberration magnitudes with increasing order. Allowing the errors to be distributed completely randomly across orders (radial frequencies) would result in unrealistic aberrations that could produce very different results.
This analysis does not directly address the question of whether aberrations are correlated in the population, nor does it rely on the existence of correlations across eyes to account for the effect described. Even in the presence of consistent correlations among aberrations in the population, it is not a priori required that real aberrations should produce better image quality than randomized aberrations. Nevertheless, there is the possibility that different eyes will show the MTF advantage effect due to qualitatively similar combinations of co-varying aberrations. This is consistent with the suggestion of Thibos, Hong et al. (2002)
that positive correlations among Zernike terms are beneficial to image quality. There tends to be a strong correlation between first-order terms (vertical and horizontal tilt) and third-order coma terms as noted by Thibos, Hong et al. (2002)
and also found in our sample; however, because the tilts do not contribute to the MTF, this relation cannot account for our results. A comparison of the correlation matrices of our data and those of Porter et al. (2001, personal communication
) to the Thibos, Hong et al. (2002)
results shows that no sets of positive correlations, but two sets of strong negative correlations were shared among all three samples, the first between terms z
(3, −3) and z
(3, −1) and the second between terms z
(4, 2), and z
(4, 4). To test whether these specific relations were important to our results, we performed another set of sign randomization simulations that preserved the signs of these four terms. The difference in the MTF ratio from the original simulation was negligible. Obviously, however, as more terms are held constant, the effect must diminish. This suggests that the MTF ratio effect is due to interactions among the entire ensemble of aberrations and not due to relations between a few pairs.
Because the sign randomization procedure preserves the RMS error of the wavefront, the MTF ratio effect suggests that the error in the true wavefronts must be distributed across the pupil differently from the error in the wavefronts with randomized-sign aberrations. To test this supposition, we sampled horizontal and vertical 2nd derivative vectors and calculated vector lengths, |D″
|, for both the true wavefronts and each randomized-aberration wavefront from Simulation 1. These samples were taken at 36 locations in concentric rings, as shown in . A short vector length (black arrow in ring 2) represents a slowly changing or relatively flatter area in the wavefront (i.e., a more planar local region, regardless of tilt), while a long vector length (white arrow in ring 4) represents a rapidly changing or less planar area in the wavefront. More rapid changes are associated with poorer optical quality. The vector lengths were arithmetically added together for each sample ring individually and for the pupil as a whole to produce a new statistic
For this measure, a ratio < 1.0 indicates that the true wavefront is flatter than the mean of the randomized wavefronts. As shown in , the true wavefronts tend to be flatter near the center of the pupil, and less flat near the pupil edge. Thus, aberrations seem to interact in real eyes to produce a wavefront shape that is relatively flatter toward the pupil center and more curved toward the pupil edge as compared to the randomized wavefronts.
Fig. 7 (A) Locations of the 36 2nd derivative samples. The white and black arrows indicate areas of rapid and slow change, respectively. The dark circles indicate rings 1–4 from the center outward. Ring 1 includes the pupil center. The white dashed circles (more ...)
Whether this wavefront shape is the result of an active, feedback-guided developmental process or just a byproduct of the physical stresses and pressures that shape the eye's optics cannot be determined from these simulations. This question could be related to the compensation between cor-neal and internal aberrations, whose mechanisms remain under debate. For example, Kelly, Mihashi, and Howland (2004)
suggests that the compensation between corneal and internal spherical aberration is a passive result of the inherent, evolutionarily determined shapes of the lens and cornea, while the values of coma and astigmatism could be developmentally fine-tuned in each eye by an active mechanism controlling lens decentration and tilt. Artal, Benito, and Tabernero (2006)
also studied the compensation of corneal and internal coma, and while they conclude that it is likely due to a passive process, they do not exclude the possibility that it is visually guided.
These simulations suggest that the Zernike components of an eye's wave aberrations are not independent and that they tend to interact in ways that produce generally flatter wavefronts toward the center of the pupil than they would if they were independent. As a result, the eye's optical quality for a given RMS error is better than it would be with independent aberrations. This effect occurs for the total aberrations of real eyes, but it was not found for corneal aberrations alone or for “synthetic” eyes, with aberrations drawn randomly from the experimental population. The relative advantage for the MTF of the true aberrations appears to occur only for larger pupils and reaches a maximum near the peak of the contrast sensitivity function.