Based on standard procedures used in optometry clinics, we compare measurements of visual acuity for 10 subjects (11 eyes tested) in the presence of natural ocular aberrations and different degrees of induced defocus, with the predictions given by a Bayesian model customized with aberrometric data of the eye. The absolute predictions of the model, without any adjustment, show good agreement with the experimental data, in terms of correlation and absolute error. The efficiency of the model is discussed in comparison with image quality metrics and other customized visual process models. An analysis of the importance and customization of each stage of the model is also given; it stresses the potential high predictive power from precise modeling of ocular and neural transfer functions.

^{2}). The room was left bright to maintain natural pupil constriction. Visual acuity was measured with a simple staircase procedure and a letter identification task. The subject was asked to name optotypes among a reduced alphabet of 18 letters, which included the usual characters in commercial charts. The letters were generated using the Sloan font developed for the Pelli–Robson contrast sensitivity chart (Pelli, Robson, & Wilkins, 1988). The test started with a letter size easily recognizable by the subject. Four random optotypes were presented consecutively, and after at least three correct answers, the tested decimal visual acuity, VA = 1/10

^{logMAR}, was increased by 1/10. The process was repeated until the subject gave less than three correct answers, at which point the tested VA was decreased by half the step. The final VA value was given by the smallest letter size for which the subject passed the 75% threshold of correct responses. All the optotypes presented to the subject were recorded for later use in the model calculations.

*M*

_{ RS}for the size of the retinal image. With the thin-lens approximation,

*M*

_{ RS}can be calculated as

*s*is the distance from the lens to the entrance pupil of the eye (it was measured to be 20 mm), and

*P*

_{ D}is the power of the ophthalmic lens in diopters. To recover the retinal visual acuity of the subject, we divided the measured acuity by

*M*

_{ RS}.

*λ*= 875 nm) LEDs mounted on the camera was used to illuminate the subject's pupil. The resolution of the system was ±0.14 mm for the measurement of the pupil diameter. The ophthalmic lenses placed in front of the pupil introduced a change in magnification of the pupil, which was taken into account by dividing the pupil diameter measurements by a factor

*M*

_{ P}in fact equal to

*M*

_{ RS}as defined in Equation 1. The mean pupil diameter measured for each letter size tested during the visual acuity measurements was stored for later use in the model computations. Hence, the fluctuation of the pupil diameter over the duration of the measurement and the effect on the optical degradation were taken into account in the model.

_{p}) to compute the retinal image. First, the ocular aberrations of the dilated pupil were reconstructed over the mean pupil diameter measured when the particular letter size was tested. The monochromatic OTF, calculated from the wavefront map, was used to recover the OTF

_{p}. The LCD used for the experiment had a broad color spectrum. Based on the literature (Llorente, Diaz-Santana, Lara-Saucedo, & Marcos, 2003), we assumed that the only ocular aberration significantly varying with wavelength in the visible is defocus, originating from the longitudinal chromatic aberration (LCA) of the human eye. As Yoon and Williams found similar effects of LCA on retinal image quality for both a broad spectrum display (a CRT in their case) and an equal energy spectrum (Yoon & Williams, 2002), we assumed a flat spectrum in our calculations. The monochromatic OTF was computed every 10 nm to cover the whole visible spectrum. The focused wavelength was taken to be 550 nm and we used the Indiana eye model (Thibos, Ye, Zhang, & Bradley, 1992) for the amount of LCA to be added at each wavelength. The effect of transverse chromatic aberration on the retinal image was ignored due to the high intersubject variability (Rynders, Lidkea, Chisholm, & Thibos, 1995) and the non-availability of data for our subjects. The sum of the monochromatic functions, weighted by the CIE standard observer photopic spectral sensitivity curve

*V*

_{λ}(Wyszecki & Stiles, 1982), gave the OTF

_{p}.

*o*

_{ i}(

**x**) is compared against an approximation

*a*

_{ i}(

**x**) of the degradation of the original optotype within each channel

*i,*i.e.,

*c*(

**x**) is the input object and {

*h*

_{ i},

**u**

_{ i}} are the modulation and shift approximating the OTF

_{p}within each channel, as given in Equation 7 in the original publication describing the model (Nestares et al., 2003). Practically, the observer finds the degradation parameters that maximizes the posterior probability

*p*({

*o*

_{i}}∣

*c,*{

*h*

_{i},

**u**

_{i}}) for each possible original optotype within each channel and selects the optotype giving the best maximum a posteriori probability (MAP). Thus, the observer implicitly estimates the OTF

_{p}sampled by the channels while estimating the optotype: This is a double Bayesian estimation. It was shown (Nestares et al., 2003) that maximizing the posterior probability for each input optotype

*j*and channel

*i*is equivalent to calculating the value

_{i}

^{j}maximizing the correlation function

*r*is the pupil radius given in millimeters,

*D*

_{ lens}is the power in diopters of the ophthalmic lens for each tested refractive state,

*D*

_{0}is the power in diopters of the lens corresponding to the peak in experimental VA, and

*cz*

_{4}is the Zernike coefficient corresponding to defocus, with the normalization given by the ANSI standards (American National Standards Institute (ANSI), 2004). After these calculations, it was found that for most subjects, the model predicted a peak in VA for a non-zero defocus,

*D*

_{0_model}in diopters, in agreement with the findings of Yoon and Williams. Therefore, assuming that the defocus state that maximizes the measured VA is the same that maximizes the predicted VA, we selected

*D*

_{0_model}to be the actual defocus at the peak of both experimental and predicted VAs. We shifted the experimental curve to have its peak located at

*D*

_{0_model}, and we performed again the model calculations to mimic the experimental conditions (pupil size variations and optotypes presented) for each refractive state tested experimentally with a new assigned defocus Zernike term

*D*

_{ lens}=

*D*

_{0}), the Zernike defocus term in the model calculations was equal to

*r*

^{2}

*D*

_{0_ model}/4√3 and hence yielded the predicted peak. The Zernike defocus terms for the other refractive states were derived from this position. This methodology resulted in having two VA curves, experimental and predicted, with matching peaks located at the defocus term that balances the other natural aberrations and maximizes the predicted VA for the stimulus spectrum.

*μ*m wavefront error RMS, except for Subject 3 OD (0.4

*μ*m RMS) and Subject 10 OD (0.61

*μ*m RMS). The error bars of the RMS plot show significant variations for some subjects, due to the changes in pupil diameter during the visual test.

*x*-axis to have its peak matched with the predicted peak, and the predictions were updated to take into account the experimental parameters associated with each refractive state (see Figure 4b). For example, the model prediction for the visual acuity at the peak of the experimental curve now includes a defocus term equal to 0.5 D. It is also based on the optotypes presented during the visual test that gave this experimental VA value and the aberration pattern reconstructed over the pupil diameter measured during that same test. The implementation of these parameters in the model is part of the approach taken to mimic tightly the experimental conditions. It can be noted that the predicted VA at 0.5 D in Figure 4b is slightly different to that for the same defocus value in Figure 4a because the latter was calculated with the experimental parameters (pupil diameter and optotypes) associated with a different VA visual test. The predicted VA curve now encompasses a different range of defocus values, and its shape is slightly altered.

*D*

_{ graph}=

*D*

_{ lens}−

*D*

_{0}+

*D*

_{0_ model}. It can be seen on the graphs that the experimental and predicted maximum of visual acuity is often at a defocus value different to zero. This is due to the fact that some amount of defocus can compensate for the other aberrations, as mentioned earlier. The points to the left of the experimental peak in visual acuity, in the hyperopic range, must be considered with caution because the subject could accommodate to compensate the effect of the ophthalmic lens, and that was not taken into account in the model. These points were removed for the subsequent analysis of the results.

*R*is 0.79. The goodness of the model fit is subject-dependant. For some subjects (e.g., Subject 5), the model predictions follow closely the measured VA, while for some others (e.g., Subjects 1 OD and 2 OS), the discrepancy is higher between the two curves. When selecting the peak VA values for each subject, the correlation between predictions and experimental data is decreased to 0.5. This finding is in agreement with a study highlighting a lack of correlation between visual acuity at the best subjective refraction and the amount of higher order aberrations (Villegas, Alcón, & Artal, 2008). In the Methods section, we detailed the procedure employed to match the peak of both experimental and model curves. It might be hypothesized that this method is not the most appropriate for some of the subjects, for whom the model curve exhibits several maxima of close values (e.g., Subjects 1 OD and 2 OS). An alternative approach, which consists in shifting the two curves to minimize the RMS error between the model and the experimental data for each subject, lowers the overall RMS error to 0.19. The last plot in Figure 5 shows the experimental and model VA group average, taken from the best focus position of each subject. The close agreement between the two curves illustrates the efficiency of the model. One might notice that the average predicted VA values are slightly lower than the measured ones. It was found that scaling the predictions by a factor of 1.03 for a least-square fit of the whole data set could only lower the overall RMS error by less than 1%. Discrepancies between the model and the experimental curves might be explained by inappropriate generic parameters, or the omission of specific features in the model. The effect and the tolerances of each particular stage of the model on the model predictions will be analyzed in the Discussion section.

*D*is the defocus value in diopters,

*D*

_{ peak}is the value of defocus where the peak in VA was obtained, and

*VA*(

*D*) is the VA at defocus

*D*. For both experimental and model curves, the calculation was performed for each subject and then averaged. With this method, the mean depth of focus was estimated to be 0.82 D for the experimental VA and 0.83 D for the model VA. Hence, the model predicts a depth of focus very close to the experimental one.

_{W}

^{−1}, the inverse of the wavefront error RMS, SFcMTF, to express the highest spatial frequency passed by the visual system, and VASmith, based on the simple relation between visual acuity and defocus, with the pupil diameter as an additional variable (Smith, 1991). The latter metric was calculated on the basis that Smith's simple relation has been used to describe through-focus visual acuity. The metrics' definitions and formulas are given in 1. The metrics were calculated with the polychromatic OTF based on the wavefront aberrations reconstructed over the median of the pupil diameter measured for each defocus value. When appropriate, the same generic neural transfer function was used as in the model calculations. The metrics were calculated for a range of defocus values in steps of 0.25 D. In the case of VASmith, it was calculated from the overall median pupil diameter and the defocus amount in diopters added from the refractive state giving maximum acuity. All the metric values were directly compared to the experimental visual acuity values. In accordance with the model readjustments, the metric predictions were shifted along the defocus axis to match their peak with the experimental VA peak.

*R*

^{2}= 0.12) for RMS

_{W}

^{−1}, a value close to that found by Villegas et al. (2008) between best-corrected high contrast VA and wavefront error RMS (

*R*

^{2}= 0.13). For the other metrics, we found even lower or not statistically significant correlation, again in agreement with Villegas et al. It should be noted that for this restricted set of experimental data, the model still performed better than the metrics (

*R*

^{2}= 0.25), showing the advantage of modeling the whole visual task for these particular conditions.

_{W}

^{−1}, SM, and SFcMTF have a mean STD

_{Focus}very close to the mean of the STD

_{Focus}of the experimental curves. The metrics STD, NS, VSMTF, and VASmith tend to underestimate the depth of focus. The relatively good performance of RMS

_{W}

^{−1}here may be explained by the fact that the aberrations are dominated by defocus for the major part of the VA curve. VASmith, on the other hand, gives poor estimation of the depth of focus as compared to other metrics. This may appear surprising since the metric is based on a relation between spherical refractive error and visual acuity (Smith, 1991). In his paper, Smith mentions that the linear relation between acuity and refractive error is not valid at low levels of refractive error; this may explain the discrepancy observed.

*μ*m. The predicted VA in these conditions shows a slightly wider depth-of-focus, although it still decreases much more rapidly than the experimental VA and than that predicted with the natural aberrations of the subject. In the context of our model predictions, it demonstrates the importance to take into account ocular aberrations, even if large amounts of defocus were added in the experiments.

*g*(

*x, y*), which is given as

*E*(Smith, 1991). Using the proportionality constant given in the cited paper, it gives for decimal VA