# Assessment of Just-Noticeable Differences for Refractive Errors and Spherical Aberration Using Visual Simulation

Purpose. The aim of this study was to evaluate the threshold levels of aberration change that a typical reference eye is able to detect.

Methods. The method involved simulation of the foveal vision of a typical eye in polychromatic light through optics affected by different levels of the various chosen monochromatic aberrations. The reference eye had the following monochromatic wavefront characteristics based on the aberrations of a population of young adults: no spherical defocus, astigmatism −0.37 D oriented at 0°, coma −0.17 D/mm oriented at 270°, and spherical aberration −0.12 D/mm^{2}. Average amounts of longitudinal and transverse chromatic aberration were assumed, and allowance was made for the Stiles-Crawford effect. The pupil diameter of the simulated eye was kept fixed at 6 mm. Three observers each compared, 100 times, a simulated image as seen through the standard reference eye with a variant “aberrated” image. The varying parameter was the value of a chosen additional aberration affecting the variant image in the reference eye. The test was repeated for varying amounts of spherical defocus, astigmatic defocus, and spherical aberration. For each of these aberrations and each observer, the discrimination probability as a function of the aberration level in the variant image was determined. The just-noticeable difference in aberration (JNDA) was derived from each discrimination curve as the difference between the aberrations corresponding to discrimination probabilities of 75% and 25%. The JNDA values obtained were expressed in the form of root mean square (RMS) wavefront error thresholds.

Results. It was found that 0.04 μm of RMS aberration should be considered as the threshold of just-noticeable image change, in good agreement with the Maréchal criterion.

Conclusions. The results imply that in normal viewing conditions (e.g., a 3-mm pupil size), optical corrections should be in the range of ±0.15 D in sphere and cylinder from the target prescription if perceptible change in the quality of the perceived images is to be avoided. The design of conventional soft contact lenses of high negative power or positive power should aim to produce −0.07 D/mm^{2} of spherical aberration, with a tolerated interval between −0.15 to +0.01 D/mm^{2} for a 6-mm pupil size.

Université Paris Sud, Laboratoire Aimé Cotton, Orsay, France (RL), Imagine Optic, Orsay, France (NC), and Department of Optometry and Neuroscience, UMIST, Manchester, United Kingdom (WNC)

Received November 21, 2003; accepted April 7, 2004.

When designing and optimizing ophthalmic corrections or laser surgery ablation procedures, it would be desirable to know the maximum wavefront error that can be tolerated by the visual system without noticeable degradation occurring in the image. The same information is needed in the optical metrology of visual instruments and ophthalmic lenses, for which the wavefront quality has to be checked against a predefined acceptance threshold. To help to define such a threshold on a visual basis, the present study aimed to measure the smallest changes in some of the primary optical aberrations that can be noticed by a human eye.

To examine the effects of aberration on visual performance when an optical device is directly coupled with the eye (e.g., a visual instrument or a contact lens), two avenues can be used.

The first method consists of having observers look through aberrated optics whose aberration levels are well known. The advantage of this method is that it simulates real life and allows an assessment of the real interaction between the optical system and the eye. In such a method, the ocular accommodation must be well measured and controlled; otherwise the results may be confounded by additional unwanted defocus effects and accommodation-dependent changes in the higher-order aberrations.^{1, 2} One major difficulty is the need to produce several specially designed optical systems if one wants to explore the effect of different levels of a single aberration or combinations of aberration.

The second method consists of simulating on a video monitor the degradation caused by the aberrations and showing these images to observers without any visual aid. The advantages of this method are the disadvantages of the first and vice versa.

Both methods have been used in the literature. The first has been used to explore aberration tolerances for visual instruments such as binoculars and telescopes.^{1, 3} The second method, presentation of computerized images on a video monitor, has also been used for the same purpose.^{4, 5} To evaluate the effect on an eye of different designs of optical device, such as intraocular lenses^{6} or contact lenses, the second method appears to be more flexible and appropriate.

Burton and Haig^{4, 5} aimed to link wavefront distortion to human visual discrimination with a view to devise a useful and reliable image quality metric. The image quality was judged in terms of the just-noticeable difference (JND) between an original image and an aberrated variant. The discrimination levels were measured using side-by-side presentations on a video monitor of the original and aberrated targets. The subjects had to judge which of the two targets appeared to have the lower level of aberration. Small, 2-mm pupils were used so that the subjects' eyes could be assumed to be diffraction limited. Haig and Burton^{5} extended the results to include higher-order spherical aberration and coma. The original images were two real targets (a scale model vehicle and a photographic print of an aerial view of a cottage) and two computer-generated targets (a random dot pattern and a black circular disk). Just-noticeable levels of individual or combined aberrations were derived from psychometric functions in the form of plots of “percent correct discrimination” against “aberration level” and were measured for 60%, 75%, and 90% discrimination probabilities.

It can be seen (Table 1) that they typically found that the just-detectable wavefront aberration lay between 0.2 and 0.4 wavelengths, about the same magnitude as the Rayleigh quarter-wavelength criterion. Table 1 also gives their results in equivalent Zernike and optometric (dioptric) terms.

We have developed a method to simulate the foveal vision of any subject's eye through optics affected by various monochromatic aberrations.^{7} The method is more general than that used by Burton and Haig, whose simulation was confined to a basically diffraction-limited eye with a 2-mm pupil. The major relevant features of our model will be summarized here. First, the wavefront aberrations of the subject's eye (in the present study, the “subject” was a “reference eye,” designed to be typical of young, real eyes) were measured. We then calculated the polychromatic point spread function of the subject's eye (PSFS) using a software program under MATLAB (Mathworks, Natick, MA) by using either the measured monochromatic aberrations (individual aberrations) or average data for a given population. Typical amounts of ocular longitudinal and chromatic aberration were included in the calculation of the PSFS, with suitable weighting for the spectral characteristics of the monitor used to display the final images (see Legras et al.^{7} for details). A second similar point-spread function (PSFA) was calculated for the subject's eye with chosen levels of additional aberration, corresponding perhaps to those of a new design of correcting lens or telescope. We convolved the target image (It) by either the calculated PSFA or PSFS to obtain a simulated retinal image (Ir) for the subject, respectively with or without the additional aberration. However, when an observer views this image on the display monitor, the Ir is further degraded by the aberrations of the observer's eye. Thus, the simulator should not show this initial calculated Ir, but rather the image (Im) that can produce Ir on the observer's retina. Im can be obtained by deconvolution of Ir by the polychromatic point-spread function (PSFO) of the observer. The complete method in the case when no aberration has been added is represented in Fig. 1. The deconvolution was computed in spatial frequency space using a point spread function Wiener filtering method.^{8} This approach was limited to cases for which the optical transfer function of the observer's eye was superior to that of the aberrated subject's eye at spatial frequencies within the bandwidth of the visual system.

The aim of the present study was to use this simulation method to evaluate the threshold levels of individual or combined aberrations that a typical reference eye is able to detect to confirm and amplify the earlier results of Burton and Haig.^{4, 5} The observer compared an image seen through a reference eye having typical amounts of ocular aberration with a variant “aberrated” image. The varying parameter was the value of a chosen aberration affecting the variant image.

## METHODS

### Psychophysical Method

A method of constant stimuli was used. The discrimination levels were measured using side-by-side presentations of an original image and an aberrated variant image involving PSFS and PSFA, respectively. The relative positions of the two images were varied randomly. The observer had to decide which was the clearer image. A forced-choice method was used; in other words, the response “equal” was not accepted. Each “aberrated” image was compared 100 times with the original image.

### Apparatus

The It was a letter E corresponding to 0.4 log min arc of visual acuity (6/15 Snellen equivalent). The two targets were generated on a monitor by a Cambridge Research System VSG3 card (Rochester, Kent, U.K.) and were viewed by the subject at a distance of 2.2 m. The observer indicated, by activating the buttons of a command box connected to the image processor, which of the two targets appeared clearer. The target separation was 1.22°, and the display luminance was 100 cd/m^{2}. Room lighting was varied to control the observer's pupil size, which was monitored during all the experiments with a CCD camera.

### Image Calculation

The original image was obtained by convolving the E-letter image with a filter corresponding to a typical reference eye (i.e., having monochromatic and chromatic aberrations corresponding to the average of a young, adult population) and then deconvolving with a filter corresponding to the observer's eye (with its individually measured monochromatic aberrations) as shown in Fig. 1. The filters were calculated in a matrix of 512 × 512 pixels with an angular distance between two pixels (θ_{p}) of 4.57.10^{−5} rad (0.157 min arc). Because we wanted to simulate the image of a Snellen letter corresponding to 0.4 log min arc of visual acuity (angular size of detail of 2.5 min arc or 7.31.10^{−4} rad), the size, in pixels, of the detail of the object was the ratio between the angular size of the detail (visual acuity) and the angular distance between two pixels of the filter. The final size of the Snellen letter was 5 and 4 times the detail size in height and width, respectively. Thus, the size of the detail of the letter in pixels was

and the letter size was 80 × 64 pixels. Around this matrix (80 × 64 pixels), white pixels were added to yield a total matrix of 512 × 512 pixels, corresponding to the required size of the filter (i.e., a total subtense of 1.34 × 1.34°).

Table 2 shows details of the filters. Because longitudinal chromatic aberration varies little between subjects, typical values were taken from the literature.^{9} Typical amounts of transverse chromatic aberration were also assumed,^{10} and allowance was made for the Stiles-Crawford effect.^{11}

To determine the typical monochromatic aberration levels for the chosen “reference eye,” as listed under “Convolution filter” in Table 2, the monochromatic aberrations of 23 young adult, right eyes were measured in a separate study using a Hartmann-Shack aberrometer. The typical levels of astigmatism, spherical aberration, and coma were derived from these data and used to determine the convolution filter for the typical reference eye.

The subjects involved in this study were aged between 20 and 40 years. The Hartmann-Shack measurements were performed with their natural pupil size (mean = 6.12 mm; SD = 0.50 mm) at the same accommodation level as that used in the main study. Pupil diameters were directly measured from the Hartmann-Shack records.

From the wavefront measurement, Zernike polynomials were fitted, and the Zernike coefficients were then transformed into optometric coefficients (i.e., corrections) expressed in diopter (D) units.

Fig. 2A shows the distribution of the fourth-order (primary) spherical aberration, expressed in D/mm^{2}, among the 23 subjects. The average spherical aberration was −0.124 D/mm^{2}. Fig. 2B shows the value and orientation of the coma aberration. The average of the absolute values is −0.174 D/mm. To determine the average orientation of the coma, we averaged the vectors and finally obtained 270°. Fig. 2C shows the values and orientation of astigmatism (expressed as negative corrections). The average of the absolute values of astigmatism is −0.384 D. To determine the average orientation of the astigmatism, we averaged the vectors and finally obtained 168°. To simplify, we decided to put in our model an astigmatism of −0.38 D oriented at 0°.

The levels of individual monochromatic aberration considered in calculating the deconvolution filters for the three individual observers who carried out the discrimination study were the coefficients of the Zernike polynomials, which fitted the wavefront measured with the same Hartmann-Shack setup.

The “aberrated” images were calculated in the same manner as the original image, except that one aberration was added to the original wavefront. We added spherical defocus ranging from −0.40 D to +0.20 D in steps of 0.025 D when testing defocus tolerance, spherical aberration ranging from −0.50 D/mm^{2} to +0.30 D/mm^{2} in steps of 0.02 D/mm^{2} when testing spherical aberration tolerance, or astigmatism ranging from −1 D to +0.40 D at 0° in steps of 0.035 D when testing astigmatism tolerance. The step chosen for each aberration tested corresponded to a change of 0.03 μm in root mean square (RMS) wavefront error.

For each aberration condition, the spherical defocus term was adjusted to maximize the modulation transfer function (MTF)-a (i.e., the area under the MTF when plotted in linear coordinates) calculated between 5 and 24 cpd. The boundary frequencies of the MTFa calculation were chosen according to the results of Mouroulis and Zhang,^{12} who showed a good correlation between subjective performance and MTFa calculated over this spatial frequency band. Guirao and Williams^{13} also found that the MTFa is an adequate metric for image quality and allows the prediction of subjective refraction.

To avoid artifacts in the displayed image, which may occur when the convolution filter corresponds to a higher optical quality than that of the deconvolution filter, we calculated the MTFs corresponding to the best aberration condition filter for the reference eye and also to the filters of deconvolution of the three observers involved in this study. If the ratio of the convolution filter over any deconvolution filter is never >1, there should be no artifacts in the displayed image. Fig. 3 shows the ratio of the best-simulated MTF over the MTFs for the three observers used in the study. It can be seen that, within the visual bandwidth up to 50 cpd, this ratio was never >1, ensuring that no artifact should be present in any of the displayed images. It is evident that to avoid artifacts the MTF of any observer's eye should be as high as possible. Therefore, we adjusted and controlled the luminance condition for each observer to obtain a pupil size for which the balance between diffraction and aberration yields an optimal MTF (i.e., pupil diameters between 2.5 and 4 mm were used^{14}). Note that because the monitor displaying the aberrated and comparison images was at a fixed distance, accommodation change was not a problem.

### Observers

Three observers were enrolled in this study. Five Hartmann-Shack wavefront measurements were made for each observer. The Zernike polynomial coefficients were averaged. Table 3 shows the main measured aberrations derived in optometric units. Aberrations up to the Zernike seventh order were considered in the calculations.

### Measurement of the Discrimination Probability Curves

The observer's task was to indicate which of the two side-by-side images appeared clearer. One hundred trials were made at each aberration setting; the relative positions of the original and aberrated image were randomized, allowing a discrimination probability to be determined. The discrimination probability was plotted as a function of each tested aberration. A percentage <50% indicated that the aberrated variant appeared clearer than the original image. A linear regression was performed on the range of probability between 25% and 75% around the zero added aberration condition (i.e., the right parts of Fig. 4). We calculated from this regression the levels of the tested aberration that corresponded to 25% and 75% probabilities. The just-noticeable difference in aberration (JNDA) was the difference divided by 2 between these two values. For errors of focus, the depth of focus (DOF) was defined as the range in diopters over which a probability of ≤75% was obtained. Similarly, for astigmatism and spherical aberration, a range of “acceptable level of aberration” was determined in the same manner as for the DOF. An optimal value of each aberration term (i.e., a balancing condition) was derived from the curves. We considered that the best balancing condition was obtained at the middle of the tolerance interval.

## RESULTS

### Defocus

Fig. 4A shows the discrimination probability as a function of the defocus aberration term for the three observers. The abscissa value corresponds to the added defocus. Table 4 shows measurements derived from the discrimination probability curves. The measured JNDA's ranged from 0.024 D (GH) to 0.048 D (MAB). The average of the three observers was 0.035 D. The DOF's ranged from 0.21 D (RL) to 0.32 D (GH) with an average of 0.28 D. The average value of optimal defocus was −0.11 D. No important differences in DOF or optimal sphere were found between the observers.

### Spherical Aberration

Fig. 4B shows the discrimination probability as a function of the spherical aberration term. The abscissa is the spherical aberration term added. Because the reference eye had −0.124 D/mm^{2} of spherical aberration (expressed in terms of correction), an added spherical aberration (the corresponding correction) of −0.12 D/mm^{2} would be expected to provide an aberration-free system (eye + optics). The curves are similar except for observer GH in the range of spherical aberration between −0.16 and −0.32 D/mm^{2}. The measured JNDA's ranged from 0.018 D/mm^{2} (RL) to 0.033 D/mm^{2} (MAB; see Table 5). The tolerated spherical aberration measured for 75% discrimination probability varied from 0.14 D/mm^{2} (RL and MAB) to 0.20 D/mm^{2} (GH). The clearest images were obtained with an added spherical aberration of −0.04, −0.06, and between −0.06 and −0.08 D/mm^{2} for observers RL, GH, and MAB, respectively, corresponding to an average value of about −0.07 D/mm^{2}.

### Astigmatism

Fig. 4C shows the discrimination probability as a function of the astigmatism term. The abscissa is the added astigmatism. An added astigmatism (correction) of −0.38 D should provide an astigmatism-free system (eye + optics). The forms of the individual curves are comparable, except for observer GH. Unlike the other two observer's curves, GH's discrimination probabilities increase slowly as the added astigmatism becomes more negative. The measured JNDA's ranged from 0.023 D (RL) to 0.056 D (GH; see Table 6). The tolerated astigmatism level measured for 75% discrimination probability varied from 0.64 D (RL) to 0.76 D (GH). The clearest images were obtained between −0.455 D and −0.07 D for observer RL and between −0.245 D and −0.175 D for observer MAB. Only one optimum added astigmatism value was found for observer GH, at −0.175 D.

## DISCUSSION

### Sources of Interobserver Differences

Even when no large difference in the shape of the curves was found between the observers, the numerical values deduced from these curves showed slight differences. The JNDA's of observer RL were often lower than those of both other observers. The JNDA's must depend on optical and neural factors. Because the optical images projected on the retina of all the observers were presumably similar, these differences in JNDA might be the result of neural differences. The following method was used to explore this possibility.

Contrast sensitivity is limited by optical and neural factors.^{15} The neural contrast threshold, represented by C_{n}, is the minimum retinal contrast that can be detected by the neurosensory system of the eye. When a spatial grating stimulus is imaged through the eye, the ratio between the retinal image contrast and the original object contrast is given by the value of MTF at the grating spatial frequency. Therefore, the contrast of the external object grating measured at the detection threshold, represented by C_{d}, is given by the following equation:

Because the contrast sensitivity (CS) is the reciprocal of the threshold contrast, we obtain:

and then, by taking logs, we obtain:

Using a staircase method, the monocular CS functions of the three observers were measured in the same room luminance conditions and at the same observation distance as was used during the JNDA experiment. The targets were sinusoidal gratings oriented vertically. The tested spatial frequencies were 1, 2, 4, 8, and 16 cpd. Fig. 5A shows the CS functions (CSF_{d}) of the three observers. The MTFs of the eyes of the observers for the appropriate pupil diameters were computed from their Hartmann-Shack measurements. For each observer, we calculated their logCSF_{n} by subtracting the log of the MTF from the logCSF_{d}. Fig. 5B shows the resultant neural CS functions (CSF_{n}) of the three observers. Observer RL shows a better CSF_{n} than the two other observers. This better CSF_{n} could suggest that this observer tolerated less image degradation, inducing lower values of JNDA. However, the small differences in CSF_{n} between observers GH and RL cannot explain the relatively large differences found in their JNDA's, particularly for astigmatism (Table 6). However, it should be noted that subject RL was more experienced. Thus, the measured differences in CSF_{n} can explain the large difference between the results obtained with subjects MAB and RL but not the differences found between RL and GH. A possible explanation could be a difference in the criteria used by the individual observers to estimate the quality of images.

### Comparison of the Results with Those of Burton and Haig and Other Studies

The measured JNDA's were transformed into Zernike coefficients (μm) to compare values independent of pupil size. Table 7 shows these JNDA's and those of Burton and Haig.^{4, 5} Regarding defocus, except for observer MAB, their results and ours were similar. We found that on average a degradation of 0.047 μm of RMS aberration can be tolerated. The tolerance to spherical aberration was on average 0.035 μm, with no significant interobserver differences. Burton and Haig found a tolerance four times less than ours. Regarding astigmatism, Burton and Haig found a tolerance similar to our result, which was 0.036 μm. In summary, except for the spherical aberration, our results were similar to those of Burton and Haig, even though different comparison images were used and an aberrated reference eye was used in our case rather than the diffraction-limited eye of the earlier study.

Applegate et al.^{16} measured visual acuity on aberrated letter charts generated using the CTView program (Sarver & Associates, Celebration, FL). The authors showed that a difference of 0.05 μm RMS of aberration significantly decreased the visual acuity. This result is in accordance with ours because the observer should obviously detect a degradation of the viewing image that involved a reduced visual acuity.

### Comparison with Maréchal Criterion

It should be noticed that whatever the aberration considered, the visual system seemed to be sensitive to a degradation of about 0.04 μm, with slight differences between subjects from 0.030 μm (RL) to 0.048 μm (MAB). This result is important because it may allow us to extrapolate the same RMS threshold to other types of aberration or to combined aberration modes. Maréchal^{17} suggested that to avoid a just-noticeable image degradation from diffraction-limited image quality, the RMS deviation of the wavefront should be <λ/14. Assuming that our reference wavelength was 550 nm, the Maréchal criterion becomes RMS <0.039 μm. Our JNDA's defined in terms of Zernike coefficients can be directly compared with the Maréchal criterion because the coefficients represent the RMS deviation of the wavefront. The average coefficient for the defocus was 0.045 μm (∼λ/12), which is slightly higher than the Maréchal criterion. For the astigmatism and spherical aberration, the average values were 0.035 μm (∼λ/16) and 0.036 μm (∼λ/15), respectively. Therefore, our results are in close agreement with the Maréchal criterion, even though the basis for their determination differs from that used by Maréchal.

### Defocus

Using the discrimination probability curves, we evaluated the DOF as the range over which a discrimination probability of ≤75% is obtained. In other words, outside this DOF a subject with a 6-mm pupil, with typical amounts of ocular aberration, can detect a visual degradation of the observed image. This definition for the DOF allows comparison of the present results with those in the literature. In our experiments, the average DOF was 0.28 D (see Table 4).

Campbell^{18} evaluated the monocular DOF of one subject by considering the range of focusing error over which no blur is perceptible. He found a DOF of 0.36 D with a 6-mm pupil size. Atchison et al.^{19} measured the monocular DOF of nine subjects aged 20 to 58 years. The target was an E-Snellen letter of 0.3 logarithm of the minimum angle of resolution. The DOF was defined as the range of focusing errors for which no perceptible change of the image was noticeable. They obtained an average DOF of 0.55 D with a pupil size of 6 mm. Our evaluated DOF was similar to the Campbell's DOF and slightly less than the one of Atchison et al. This difference could be the result of the different criteria and methodology used to define the DOF: Campbell's method was essentially a comparison method like ours, whereas this was not the case with the method used by Atchison et al.

An added defocus of −0.11 D was subjectively preferred to the sphere of the original image (see Fig. 4A). As noted earlier, we optimized the sphere by calculating the MTFa between 5 to 24 cpd. This range was based on the conclusion of Mouroulis and Zhang,^{12} which was that this metric was well correlated to subjective discrimination. Optimization of the sphere by calculating the MTFa between 5 and 15 cpd would have predicted a sphere of −0.075 D, which is closer to our value.

Any choice of the metrics used for optimization necessarily involves somewhat arbitrary assumptions. It is probable that, in the presence of spherical aberration, the optimal sphere depends somewhat on the type of test object used.^{20}

### Spherical Aberration

The subjectively preferred spherical aberration level was −0.07 D/mm^{2} on average (see Table 5). In relation to our average reference eye model, this may suggest that it is not desirable to fully correct the spherical aberration.

It is interesting to relate this finding to the contact lens literature. Chateau et al.^{21} tested soft contact lenses with different levels of spherical aberration in two samples of subjects aged 20 to 45 years (18 emmetropes and 19 myopes). CS was measured at 12 cpd to determine the optimal lens spherical aberration required by each individual. Chateau et al. found that, for the emmetropic group, which corresponds better to the population chosen to determine the average aberration levels of our eye model, the lens with a spherical aberration coefficient of −0.074 D/mm^{2} provided better CS than that with −0.14 D/mm^{2}. Thus, if we consider that the average spherical aberration level of the subjects involved in Chateau's study was comparable with our average level (−0.12 D/mm^{2}), the result was in agreement with that of our study. However, it must be remembered that any soft contact lens designed to compensate for the spherical aberration of the eye should be well centered. If not, the negative effects of other aberrations induced by the decentration may counterbalance the enhancement in visual performance provided by this correction. In this situation, the fully corrective contact lens may not give the best visual performance.

Our results show that an eye with a typical amount of spherical aberration will not detect degradation in the range of ±0.08 D/mm^{2} around the optimal value (−0.07 D/mm^{2}). In other words, an optical system with spherical aberration between −0.15 and +0.01 D/mm^{2} will not cause perceptible degradation.

Our Hartmann-Shack aberrometer measurements, like those of Cheng et al.,^{22} showed that spherical aberration is essentially independent of ametropia in the range from −8 to +4 D. Results for a larger sample of 50 young adult eyes are shown in Fig. 6. The average spherical aberration coefficient is −0.12 D/mm^{2}, which is the value we have chosen for our reference eye.

In conventional spherical contact lenses, spherical aberration results from the spherical shapes of the anterior and posterior optical surfaces. Although conventional spherical contact lenses of negative dioptric power, say <8 D, exhibit spherical aberration <−0.15 D/mm^{2}, the limit of our tolerance, the higher aberration levels occurring in negative lenses of higher power and in positive lenses will be detectable as a degradation by the visual system. Thus, conventional soft contact lenses of high negative or positive powers should be corrected for their spherical aberration. Ametropic wearers of these lenses will then obtain a better level of visual comfort.

Because the mean JND of spherical aberration was 0.024 D/mm^{2} (see Table 5), a −0.07 D/mm^{2} correction of spherical aberration will be detectable and may provide visual benefit. In real life, rather than under the controlled conditions of our study, the JND of spherical aberration may be higher, and correction may only provide a visual benefit during critical visual tasks.

### Astigmatism

The average subjectively preferred cylinder of astigmatism was about −0.31 D (see Table 6). This was not a surprising result because the astigmatism (correction required) of our reference eye model was −0.38 D.

The tolerance to astigmatism obtained from our JNDA evaluation was 0.035 μm (Table 7). For a 3-mm pupil size, this means that an individual will tolerate an uncorrected cylindrical refractive error of 0.15 D. Charman and Voisin^{23} argued that under typical observing conditions, 0.25 D of uncorrected cylindrical refractive error is unlikely to produce any appreciable loss in the optical image quality at the retina but that the use of a more conservative tolerance value (e.g., 0.15 D) may well be justified in some circumstances. Our result is in accordance with the conclusion of Charman and Voisin because the visual task used in our experiment was highly discriminant.

### Practical Application to Contact Lens Tolerances

Physiologic tolerance data against typical aberrations should help the manufacturers when inspecting their contact lenses. Unfortunately, the manufacture of soft contact lenses suffers from process deviation, and it is not unusual to find large (up to 0.50 D) errors in spherical power or residual astigmatism even in spherical lenses. Our data suggest that in normal viewing conditions (e.g., a 3-mm pupil size), the power of the contact lens should be within the range of ±0.15 D in sphere and cylinder from its nominal value. Conventional soft contact lenses of high negative power or positive power should also be compensated for their spherical aberration, targeting a final value of −0.07 D/mm^{2}.

### Recommendations for Customized Corrections

Advances in the measurement of the eye's wave aberration and the ability to correct it with customized corrections raise the possibility of correcting the aberrations. Do the visual benefits of correcting all the aberrations warrant customized laser refractive surgery or customized intraocular lenses and contact lenses?

Insofar as the characteristics of our reference eye are representative of typical levels of ocular aberration, it is evident from Fig. 4 and Tables 4 through 6 that refining the spherocylindrical correction and correcting spherical aberration can yield a noticeably better optical image when the pupil diameter is 6 mm. Do the experimental results have wider application?

The evaluated JNDA's could serve as a baseline to decide whether to correct any measured level of aberration. When correcting aberrations with customized corrections, one should keep in mind that visual performance, such as visual acuity, is affected not only by the measured monochromatic aberrations but also by temporal changes in the monochromatic aberration, by neural resolution, by the continuing presence of chromatic aberration, and by others factors.^{24} Even in the presence of these factors, we are able to detect a small change of monochromatic aberrations. Table 7 suggests that any aberration less than about 0.04 μm is not worth correcting because its correction will not be detectable. It should again be remembered that this criterion has been obtained in highly controlled conditions and using a highly discriminant visual task. In natural life, the tolerable aberration level will probably be higher. With >0.04 μm aberration, a customized correction could be considered. But at present, it may be doubted whether customized corrections can routinely achieve a sufficient level of accuracy.

## ACKNOWLEDGMENTS

We thank Bruno Fermigier for providing the image calculation software.

Richard Legras

Université Paris Sud

Laboratoire Aimé Cotton

Campus d'Orsay

Bât. 50591405

Orsay Cedex

France

e-mail: richard.legras@lac.u-psud.fr

## REFERENCES

**Keywords:**

simulation; aberrations; tolerances; defocus; spherical aberration; astigmatism