1. Introduction

Chromatic aberration occurs because of the dispersion of optical media, where for typical materials the refractive index decreases with increasing wavelength. Longitudinal chromatic aberration (LCA – also known as axial chromatic aberration) occurs when different wavelengths of light focus at different distances from the lens. Transverse chromatic aberration (TCA – also known as lateral chromatic aberration) occurs when different wavelengths of light focus at different positions in the image plane (Fig. 1). As a result of chromatic aberration, images contain so-called fringes of color and the image quality is degraded.

 

Fig. 1. LCA (left) and TCA (right).

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Going back to Thomas Young in 1802 [1], various authors have demonstrated that the human eye exhibits LCA at the fovea. Studies where authors measured at least four subjects indicate that all eyes exhibit a significant amount of foveal LCA, ranging from approximately 1.5 to 3.9 D when comparing 700 to 400 nm light [2–9]. It has also been shown that, although some subjects exhibit small amounts of foveal TCA, others exhibit a substantial amount, with the amount ranging from approximately 0 to 3 arcminutes when comparing 700 to 400 nm light [9–11]. Thus, there is significant foveal LCA in all eyes, there is significant foveal TCA in many eyes, and the amount of LCA and TCA in the eye varies considerably across subjects.

Although retinal image quality in the fovea is significantly degraded by chromatic aberration, does visual performance improve when chromatic aberration is corrected? Evaluation of the impact of chromatic aberrations on visual performance has been conducted in two ways. In one approach, performance with spectrally narrowband stimuli was compared to performance with broadband stimuli. In the other approach, an achromatizing lens was used to correct LCA and performance was compared to performance without the achromatizing lens. It has been shown that grating acuity [12] and contrast sensitivity [13,14] are improved in monochromatic conditions when compared to polychromatic conditions. However, correction of LCA has produced mixed results, some demonstrating no improvement in contrast sensitivity [13] or a degradation in visual acuity (VA) [8] and others demonstrating a small improvement in contrast sensitivity [15]. Given that the literature presents conflicting results on the impact of correction of chromatic aberrations, the first author and colleagues endeavored to carefully measure and correct both LCA and TCA on an individual subject basis [9]. They showed that correcting chromatic aberrations resulted in a 0.2- to 0.8-line improvement in VA. Here, we ask, is this measured improvement consistent with what we would have expected from an analysis of the effect of chromatic aberration on the information available to the visual system for the letter acuity task?

2. Methods

In this paper, we characterize the theoretical impact of chromatic aberration correction on visual acuity. We did this in three steps: (1) we developed tools to simulate the optical impact of physiologically relevant amounts of LCA and TCA in real human eyes, (2) we developed tools to simulate VA in an ideal observer, (3) we employed these tools to characterize the impact of chromatic aberration on VA. That is, we model how the initial stages of the human visual system encode information, how an ideal observer uses this information to make perceptual decisions, and how ideal observer VA depends on LCA and TCA.

The optics of the eye relay information from the object (the distal stimulus) to an image available at the retina (the proximal stimulus). The retinal photoreceptor cone mosaic encodes that information through cone photopigment excitations, which are subject to unavoidable Poisson noise. The visual system then uses the information to make an inference on each trial of a simulated psychophysical task that probes letter acuity. We model the inference as an ideal observer with access to the cone excitations [16], so that our computed thresholds represent the performance of a statistically optimal observer limited by the Poisson noise in those excitations.

2.1 Optical model of chromatic aberration

To generate physiologically relevant polychromatic eye models, we employ data from the literature to estimate population variation. We employ wavefront characterizations of the eyes’ monochromatic aberrations and combine these with LCA and TCA measurements from the literature, to bound and define the model eye parameters. Measurements of LCA and TCA in the literature were acquired at two wavelengths in the visible spectrum, but we need to model the full wavelength dependence of the LCA and TCA. We used reduced-eye models from the literature to estimate LCA and TCA as a function of wavelength, and combined this with the monochromatic wavefront characterization to generate 882 polychromatic model eyes (18 subjects; 49 LCA/TCA combinations per subject) as described in detail below.

We used optical principles to estimate the LCA and/or TCA over wavelengths spanning the visible spectrum, from measurements at just a few wavelengths, following eye models whose mathematical descriptions were provided previously [9]. Briefly, we use a version of Emsley’s reduced-eye model [17] that was modified by Thibos et al. [5] to model LCA. This was achieved by attributing variation in LCA to individual differences in dispersion of the ocular media. This led to a system of equations that, with inputs of the chromatic difference of focus of three wavelengths, can be solved to generate chromatic difference of focus curves representative of various observers [9].

We use a version of Gullstrand’s reduced-eye model [18] that was modified by Thibos [19] to model TCA. This was achieved by attributing variation in TCA to individual differences in the angle between the visual and optical axis of the eye. We derived a transcendental equation that, with inputs of the chromatic difference of the chief ray angle at three wavelengths, can be solved numerically to generate chromatic difference of chief ray angle curves (as a function of wavelength) representative of various observers [9].

With these principles, we modeled the LCA and TCA for a range of values chosen to represent the range observed experimentally. We set the chromatic difference of focus and the chromatic difference of chief ray angle to zero at 555 nm. In total, 7 values of LCA (0, 1.278, 1.740, 2.203, 2.665, 3.128, and 3.590 D from 700 to 400 nm) and 7 values of TCA (paired, vertical: 0, 0.40, 1.58, 2.76, 3.94, 5.12, and 6.30 arcmin and horizontal: 0, 0.20, 0.79, 1.38, 1.97, 2.56, and 3.15 arcmin from 700 to 400 nm) were simulated (Fig. 2). A description of the method used to determine this range of LCA and TCA values is provided in Supplement 1. Note that for the LCA, the population average is in the middle of the range simulated, whereas for TCA, the population average is at the low end of the range.

 

Fig. 2. Chromatic difference of focus as a function of wavelength for a nominally average eye, with the corresponding values at 400 nm and 700 nm shown to illustrate the LCA (top left). Chromatic difference of chief ray angle as a function of wavelength for an eye in between the population average and the maximum from the literature, with the corresponding values at 400 nm and 700 nm shown to illustrate the vertical TCA (top right). In total, 7 values of LCA (bottom left) and 7 values of vertical TCA (bottom right) were simulated (6 shown with colored lines, the seventh condition, zero chromatic aberration, is shown with a gray line).

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Simulation of optical performance was performed using 18 different monochromatic eye models. These are 18 eyes whose wavefronts (computed through 6^th order Zernike polynomials) were acquired previously in a study with 74 subjects [20]. The 18 wavefronts were drawn equally spaced from an ordered list of 74 subjects, after ordering them by their higher-order root-mean-square (RMS) aberration. The subject age range was 22–40 years; no gender information was provided. Each wavefront aberration was computed from the average of the Zernike coefficients extracted from three high-quality wavefront sensor images. The wavefront chosen to compute the point-spread function (PSF) corresponded to the wavefront with a defocus value chosen to yield the maximum monochromatic Strehl ratio. This was determined by computing the through-focus PSF and corresponding Strehl ratio in 0.1 D steps. Note that the optimum Strehl ratio does not necessarily correspond to a defocus of zero, as it would for a diffraction limited eye. Entrance pupil diameter was set to 4.0 mm for this calculation. The PSF was computed using a Fourier method [21]. PSFs were generated every 20 nm from 380 nm to 760 nm (19 PSFs per eye model) for each of the 18 monochromatic eye models with each of 7 LCA and 7 TCA models described above, giving a total of 18 × 7x7 = 882 polychromatic eye models (each described by 19 PSFs spanning the visible spectrum in 20 nm steps). An illustration of one of the resulting polychromatic eye models is shown (Fig. 3); in this case, the eye with median monochromatic optical quality, average LCA (2.2 D), and high TCA (2.76 acrmin vertical), is shown. Note that what we refer to as high TCA is not the largest value we investigated.

 

Fig. 3. PSFs at 400 nm (left column), 560 nm (middle column), and 700 nm (right column) for the eye with median monochromatic optical quality (Subject 9), average LCA (2.2 D) and high TCA (2.76 arcmin). PSFs with only monochromatic aberrations (top row), with monochromatic aberrations and LCA (middle row), and with monochromatic aberrations and LCA and TCA (bottom row). Cyan crosshair marks are included to better illustrate the impact of TCA. Note that with LCA, the PSF is much more blurred at 400 nm than at 700 nm (middle row). This is what we expect: because of the non-linear and asymmetric dependence of LCA on wavelength, violet light at 400 nm is more out of focus than red light at 700 nm. Similarly note that with TCA, the PSF is displaced more at 400 nm than at 700 nm (bottom row). Again, this is what we expect: because of the non-linear and asymmetric dependence of TCA on wavelength, violet light is offset more than red light.

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2.2 Simulation of VA with an ideal observer

Performance simulations were performed using ISETBio (isetbio.org) and the accompanying ISETBioCSFGenerator package [22]. Acuity experiments were simulated for a four alternative forced choice (4AFC) tumbling-E task. There are 4 alternatives because the letter E can be in one of four orientations: right, up, left, or down. Stimuli were achromatic (gray) with 20% contrast (Fig. 4, left column). We estimate visual acuity as the minimum size of the letter where the observer can obtain a criterion level of 78.1% correct. Background luminance was either 4 cd/m² (low luminance) or 160 cd/m² (high luminance) with spectral properties matching an Active Matrix Organic Light-Emitting Diode (AMOLED) display (Waveshare 5.5-in. HDMI AMOLED; see Fig. 5) to match the experiments in [9]. To further match the experiments, stimulus duration was simulated as 500 msec. For each polychromatic model eye, the stimulus was convolved with the PSF to determine the retinal image, and the stimulus properties were used to determine the retinal irradiance. This was done for each wavelength sampled across the visible spectrum, shown here for 400, 560 and 700 nm wavelengths (Fig. 4). Calculation of retinal irradiance took the spectrally-selective absorption of light by the crystalline lens and macular pigment into account.

 

Fig. 4. Acuity experiment stimuli. 4AFC tumbling-E, achromatic gray with 20% contrast (left column). Example retinal image (proximal stimulus) at three wavelengths, 400 (left-middle column), 500 (right-middle column), and 700 nm (right column). Retinal images were normalized individually to use the full range of the grayscale for illustrative purposes. The scale bar for each wavelength provides the retinal irradiance. These correspond to the PSFs shown in Fig. 3.

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Fig. 5. Spectral properties of the 4AFC tumbling E stimulus simulated on an AMOLED display. Shown here is the normalized spectral radiance of the stimulus.

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From the light distribution on the retina, we simulate the excitations of the mosaic of cone photoreceptors. A model cone mosaic was generated using cone density measurements from Curcio et al. [23], with cone density varying with eccentricity as described in [22]. The L- and M-cone identities were determined randomly with 2-times as many L-cones as M-cones. 7% of the cones were nominally S-cones, but the central 0.3-degrees of the fovea had no S-cones (tritanopic zone), that is, the central 0.3-degrees of the fovea had only L- and M-cones [22] (Fig. 6). Thus, the actual proportion of S-cones in the model mosaic was lower than 7%. The spectral quantum efficiency of the S-, M-, and L-cones were based on Stockman & Sharp [24,25]. We used a lens age of 32 years when computing the retinal image, and macular pigment optical density (MPOD) that varied across the mosaic with a peak of 0.35 at the fovea. An example cone mosaic is shown at the right of Fig. 6 with the L-cones labeled in red, the M-cones labeled in green, and the S-cones labeled in blue.

 

Fig. 6. L-, M-, and S-cone excitation probabilities (left, excitation probabilities shown include effects of crystalline lens and macular pigment) and an example cone mosaic with the L-cones labeled in red, the M-cones labeled in green and the S-cones labeled in blue (right).

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Given the simulated cone mosaic, ISETBio routines allow calculation of the mean number of photopigment excitations of each cone in the mosaic. The mean excitations for the full mosaic of cones describes the initial neural encoding of the visual stimulus. The variability of this encoding is modeled by a Poisson distribution for each cone, with the mean of that distribution corresponding to the mean number of excitations. In the work reported here, we analyze the level of performance possible in a psychophysical task based on this initial Poisson-limited encoding.

To compute performance in the letter acuity task for an ideal observer with access to the cone excitations, we need to model how the observer makes an inference about letter orientation. That is, we need to model how the ideal observer decides, is the E pointed to the right, up, left, or down? For this, we use the Poisson-limited ideal observer model [16] under the assumption that the observer has knowledge of the noise-free excitations to the stimulus in each orientation, for each letter size considered. This observer uses all the available encoded information to perform the psychophysical task. For a given letter size and light level, the noise-free cone excitations were calculated for each stimulus orientation. Figure 7 (left column) depicts noise-free cone modulations derived from the noise-free cone excitations. This calculation depends on the optics, cone mosaic layout, and cone fundamentals. Across the computations reported here, we held the properties of the cone mosaic and fundamentals fixed as described above. The calculations were repeated for each of the 882 polychromatic eye models.

 

Fig. 7. Example noise-free cone modulations for each stimulus orientation, for the low luminance condition, where the modulation is computed as M = (E-B)/B, where E is the excitation to the stimulus and B is the background excitation (left column). The cone modulations for each cone in the foveal mosaic were calculated using the ISETBio software. The calculations incorporate optical blur from a polychromatic eye model as well as the parameters defining the layout and spectral excitation probability for each cone in the mosaic. Example noisy cone excitations (left-middle, right-middle, and right columns). Three instances shown to illustrate the impact of Poisson noise. These correspond to the PSFs shown in Fig. 3 and are for a letter size of 0.1 degrees.

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The spatial pattern of noise-free cone excitations provides a stimulus template used by the ideal observer to make a decision on each simulated trial of the psychophysical experiment. The information available on each trial, however, is not the noise-free excitations but rather a noisy instance of those excitations. Figure 7 shows an example of the effect of this noise, in the form of three independent excitation instances for a single letter orientation. Although the orientation of the letter and noise-free excitations from the stimulus are the same across these four instances, the variability across the instances makes clear that the Poisson (also known as shot) noise can hinder determination of the letter orientation. This noise occurs because of the quantum nature of light resulting in the probabilistic absorption of photons by photopigment molecules. As a result of Poisson noise, the standard deviation of a cone's photopigment excitations is proportional to the square root of the number of excitations. In a simulated experimental trial, the templates are used to compute the likelihood that the noisy signal arose from the stimulus at each orientation, given the Poisson noise. The stimulus orientation that has the highest likelihood is then chosen by the ideal observer [16,26]. This method maximizes the observer's percent correct on the psychophysical task, given that each of the orientations was equally likely a priori. A detailed description of the ideal observer classification task for the 4AFC task is provided in Supplement 1

We simulated 512 trials for each stimulus size and polychromatic model eye, and used the ideal observer decision model to compute percent correct. A total of 9 stimulus sizes were simulated for each polychromatic model eye, with a minimum letter size of 0.002 degrees and a maximum letter size of 0.2 degrees. The size of the stimulus on each block of 512 trials was determined adaptively using the QUEST + method [27,28]. The slope of the psychometric function was allowed to vary from 0.05 to 25. After the simulated session of 9 stimulus sizes, percent correct was plotted as a function of stimulus size and a Weibull function was fit to the data. Threshold was calculated from the fit as the stimulus size where 78.1% of trials are correct (Figs. 8 and S1). To convert to VA, we find the Minimum Angle of Resolution (MAR) in arcminutes of a single bar of the threshold E-stimulus, such that MAR = threshold·60/5. We multiply by 60 to convert from degrees to arcminutes and we divide by 5 to convert from the full vertical stimulus subtense to the subtense of a single horizontal bar of the letter E (3 bars and 2 spaces across the full letter height).

 

Fig. 8. Example psychometric function showing percent correct as a function of tumbling-E stimulus size. Circles represent the data, the solid line represents the Weibull fit to the data, and the dotted line illustrates the threshold. The letter size of the stimulus at threshold is displayed in the plot title and corresponds to -0.108 logMAR. This psychometric function is for the subject with median optical quality (Subject 9), LCA of 2.2 D, and TCA of 0.4 arcmin vertical.

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2.3 Simulating the impact of chromatic aberration correction on VA

Visual acuity was predicted for each of the 18 monochromatic eye models with each of the 7 LCA and 7 TCA models described above, giving a total of 18 × 7x7 = 882 VA predictions, one for each of the 882 polychromatic eye models. This provided quantification of the impact of the magnitude of LCA and TCA on VA. The monochromatic Strehl ratio of each of the 18 monochromatic eye models was also calculated, and the impact of monochromatic Strehl ratio on the acuity improvement due to LCA and TCA correction was characterized.

3. Results

3.1 Low luminance VA

A set of psychometric functions are provided in Supplement 1 (Figure S1) to illustrate that they are well determined across various conditions.

In Fig. 9, we show the low luminance VA results for each of the 49 different polychromatic eye models for 3 different monochromatic eyes: Subject 1 (the subject with the best monochromatic optical quality), Subject 9 (the subject with the median monochromatic optical quality, data from this subject are also shown in Figs. 3,4,7,8 and S1), and Subject 18 (the subject with the worst monochromatic optical quality). In the panels at the left, we show the visual acuity as a function LCA in diopters. Each of the different curves represents a model eye with different amounts of TCA, as specified in the legend. First, consider Subject 9 (middle row). The results of Subject 9 show that, the more LCA the subject has, the worse the VA gets. The trend appears to be roughly linear. We also see that there is a trend where the best VA is obtained in eyes with lower amounts of TCA.

 

Fig. 9. Low luminance VA as a function of LCA for eyes with 7 different amounts of TCA (left column). Change in VA as a function of LCA correction, again, for eyes with 7 different amounts of TCA (right column). TCA values are color coded as indicated in the figure legend. Recall that TCA is the difference in chief ray angle comparing 400 and 700 nm light. Results for three subjects are shown: Subject 1 (the subject with the best monochromatic optical quality), Subject 9 (the subject with the median monochromatic optical quality), and Subject 18 (the subject with the worst monochromatic optical quality). Note that the right column shows the change in VA that we would expect for full correction of the LCA in an eye model with the amount of LCA shown on the x-axis.

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Another way of looking at these same data is shown in the panels at the right where we calculate the change in VA as a function of LCA, again with the same 7 color schemes that encode different amounts of TCA. These plots show how much VA will change when fully correcting LCA, for eye models with different amounts of TCA. For example, take an eye model with the monochromatic wavefront of Subject 9, with 2.2D of LCA and 0.4 arcmin of vertical TCA. For the ideal observer, full correction of that eye’s LCA results in a change in VA of -0.097 logMAR (Fig. 9: middle row, right column and Figure S1: middle row, middle column compared to middle row, left column). To help interpret this value, note that 0.1 logMAR is 1-line on an acuity chart, and a half-line change in VA is generally taken as clinically meaningful. Further, 0.02 logMAR is 1-letter of VA, so that this is an improvement of approximately 5-letters.

Across TCA conditions, the improvements provided by correcting LCA are reduced as the amount TCA increases. Eyes with zero and average TCA generally receive the full benefit of LCA correction, but the benefits are reduced in subjects with high TCA and further reduced in subjects with very high TCA.

The top row of Fig. 9 shows results for a subject with much better monochromatic optical quality (Subject 1). Here we see a more pronounced but otherwise similar trend, where correcting LCA improves VA, correcting TCA also improves VA, although to a lesser extent, and the impact of correcting LCA is less in eyes with more TCA. For Subject 18 (bottom row), the subject with the worst monochromatic optical quality, we again see a similar trend but here the impact of chromatic aberrations is greatly attenuated. Putting these results together, subjects with better baseline monochromatic optical quality are expected to benefit more from LCA correction, and subjects are expected to benefit more from LCA correction than from TCA correction.

In Fig. 10, we plot the change in low luminance VA as a function of the monochromatic Strehl ratio for all 18 subjects to summarize the impact of LCA correction. The Strehl ratio is a metric that describes optical quality, where a higher number is better; 1 is ideal and corresponds to a diffraction limited optical system. Consequently, points on the right side of each panel represent subjects with less monochromatic aberrations and better optics. Conversely, data on the left side of each panel represent subjects with more monochromatic aberrations and worse optics.

 

Fig. 10. Change in low luminance VA as a function of monochromatic Strehl ratio for all 18 subjects showing the impact of LCA correction for different amounts of uncorrected TCA and baseline LCA (as indicated in the panel titles). A linear fit is shown with a solid blue line, and the equation is provided in the top right of each panel. LCA is the difference of focus comparing 400 and 700 nm light and TCA is the difference in chief ray angle comparing 400 and 700 nm light.

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The top left panel of Fig. 10 shows eyes with 2.2 D of LCA and 0.4 arcmin of vertical TCA in the pre-correction condition, and the change in VA post LCA correction. Thus, the change corresponds to correction to 0 D of LCA along with the uncorrected 0.4 arcmin of vertical TCA. The improvement increases significantly with monochromatic Strehl ratio; there is more of an improvement for subjects with a higher Strehl ratio. Recall that this upper left plot corresponds to average amounts of LCA and TCA. So, in the population, we expect correction of LCA alone to provide for an improvement in VA of between approximately 0.04 and 0.16 logMAR for the ideal observer. That ranges between 0.4- to 1.6-lines of improvement, depending on the optical quality of the subject. In the panel at the bottom left of Fig. 10, we have subjects with 2.7 D of LCA and 0.4 arcmin of vertical TCA in the pre-correction condition. Note that 2.7 D of LCA is closer to the higher end of what has been observed experimentally. Here, the improvements are more significant, ranging from 0.05 to almost 0.2 logMAR. In the panel at the top right, we again start with subjects with 2.2 D of LCA, but in this instance, they have 2.76 arcmin of vertical TCA. Note that 2.76 arcmin is on the high end of what has been observed experimentally. Here, we still see improvements with correction of LCA, but the improvements are attenuated, ranging from approximately 0.04 to 0.12 logMAR. Finally, in the panel in the bottom right, with 2.7 D of LCA and 2.76 arcmin of TCA, we again see a greater improvement with correction of LCA, again because at baseline subjects had more LCA. Compared to the low TCA condition, however, the improvements are attenuated, ranging from 0.05 to 0.13 logMAR.

Considering these results together, we see that, across all four panels of Fig. 10, the change in VA is correlated with the monochromatic Strehl ratio. Thus, subjects with less monochromatic aberrations would benefit more from LCA correction. Similarly, comparing the results between the two columns, we see that subjects with less TCA would benefit more from LCA correction.

In Fig. 11, we again plot the change in low luminance VA as a function of the monochromatic Strehl ratio, but here, we illustrate the impact of correcting only TCA. In the panel at the top left, we have subjects with 2.2 D of LCA and 0.4 arcmin of vertical TCA in the pre-correction condition, and then we simulate VA post TCA correction, so in the post condition, they have 2.2 D of LCA and 0 arcmin of TCA. The change in VA is generally very small, 0.2-line at most, and the change is largely independent of the monochromatic aberrations. In the panel at the bottom left, for a subject with high amounts of LCA, we again see little impact of TCA correction. For a subject with high amounts of TCA, there is a bit more of an improvement with TCA correction, but the benefits are still small, perhaps 0.4-line at most. Lastly, the panel at the bottom right shows that, for a subject with high LCA and high TCA, we again see a modest benefit of TCA correction, about 0.2-line of improvement at most.

 

Fig. 11. Change in low luminance VA as a function of monochromatic Strehl ratio for all 18 subjects showing the impact of TCA correction for different amounts of uncorrected LCA and baseline vertical TCA (as indicated in the panel titles). A linear fit is shown with a solid blue line and the equation is provided in the top right of each panel. LCA is the difference of focus comparing 400 and 700 nm light and TCA is the difference in chief ray angle comparing 400 and 700 nm light.

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The TCA-correction trends are quite similar across the panels; none of the results show a significant impact of TCA correction, and the trend does not appear to scale with monochromatic optical quality. This suggests that, in the physiological range, and in the presence of LCA, correction of TCA has little impact on the information available for VA, perhaps improving ideal observer acuity by about 0.2- or 0.4-line.

3.2 High luminance VA

Results for the high luminance condition are presented in the Supplement (Figures S2-S4) and illustrate that the changes in VA are nearly independent of luminance. Compared with the low luminance results, the VA is significantly better (lower) in the high luminance condition. Higher VA is expected as luminance increases, because the Poisson-limited signal-to-noise ratio of the cone excitations increases with the square root of luminance. Note that the exceptionally good acuity observed in some eyes, particularly with correction of chromatic aberrations, appears to exceed the Nyquist limit. This finding is addressed in the Discussion. Generally, the trends regarding the impact of LCA and TCA are quite similar when comparing high and low luminance. That is, for the high luminance VA results, subjects with better baseline monochromatic optical quality benefit more from LCA correction, and subjects benefit more from LCA correction than from TCA correction.

4. Discussion

In this manuscript, we developed tools for simulating the impact on VA of physiologically relevant amounts of chromatic aberration in real human eyes, and we deployed these tools to evaluate how an ideal observer’s VA would be affected by correction of chromatic aberration. Notably, the calculations take monochromatic aberrations into account as part of the evaluation. Results indicate a substantive improvement of 0.4- to 2-lines in VA with chromatic aberration correction, but the benefit is not homogeneous across different levels of monochromatic aberration. In general, there is more benefit from correction of LCA than from correction of TCA and improvements in visual acuity are greater in subjects with less monochromatic aberrations, such that subjects with better baseline optical quality would benefit most from correction of chromatic aberrations. In addition, the benefit of LCA correction depends on the baseline level of TCA, such that subjects with less TCA would benefit most from correction of LCA.

A brief treatment of the role of model parameter choice on VA and the effect of chromatic aberration correction is presented in Section 4.1. It is important to note that other factors such as accommodation and eye movements may influence VA and the role of chromatic aberration, but they were not included in the simulations presented here. Those limitations are discussed in Section 4.2. In prior experimental work, the first author and colleagues showed that correcting chromatic aberrations resulted in a 0.2- to 0.8-line improvement in visual acuity in presbyopic and/or non-accommodating eyes [9]. Is this what we would have expected? In Section 4.3 we compare the findings of the current simulations with the experimental work. In addition, the results presented here indicate that the ideal observer is capable of VA that exceeds the Nyquist limit set by the cone photoreceptor spacing. We present rationale for why this result is reasonable in Section 4.4.

4.1 Role of model parameters on performance

To examine the robustness of our key findings to details of the model parameters, we conducted exploratory calculations on the role of pupil diameter, visual system integration time, lens optical density, MPOD, and L:M cone proportion on the effect of chromatic aberration correction on VA. For brevity, we present only the topline findings here. The repository that accompanies this paper provides code that performs the VA calculations described below, and data that allows further characterization by the interested reader.

On the role of pupil diameter, we first examined the variation in the monochromatic modulation transfer function (MTF) for each of the 18 monochromatic wavefronts in the study. MTF area was computed in the absence of LCA and TCA for spatial frequencies between 3 and 12 cycles per degree for pupil sizes ranging from 1.5 to 4.0 mm in 0.1 mm steps. The analysis was conducted separately for MTFs along the x- and y-directions with similar results for the two directions. We were limited to diameters equal to or less than 4 mm because the wavefronts were available for a 4 mm pupil diameter. We found that the area of the MTF was nominally quadratic with entrance pupil diameter, it peaked somewhere between the minimum possible human pupil diameter of 1.5 mm and the analysis pupil diameter of 4 mm. We expect this because, in the human eye, aberrations limit optical performance for large pupils and diffraction limits optical performance for smaller pupils. We found that the entrance pupil diameter at peak optical performance was typically between 3 and 4 mm. This is generally consistent with the literature on the topic [29].

Next, we examined VA and VA improvement with correction of chromatic aberration for pupil diameters between 1.5 and 4 mm in 0.5 mm steps. We did this only for Subject 9, the subject with median RMS wavefront. We found that VA decreases with decreasing pupil diameter. We expect this because decreasing retinal illuminance reduces the signal-to-noise ratio (SNR) of the information carried by the Poisson noise limited cone excitations. Improvements afforded by correction of chromatic aberration were greatest for the 4.0 and 3.5 mm pupil diameters and were then reduced with decreasing pupil size. This is expected given our across subjects analysis, where the benefits of correction of chromatic aberration are smallest for eyes with the lowest monochromatic optical quality. For a 1.5 mm pupil, the improvement in VA across all the LCA and TCA conditions we examined was less than 0.04 logMAR.

With respect to other parameters, again for Subject 9, we characterized VA with: a 100 ms integration time, MPOD of 0.7, lens age of 60 years, a deuteranopic cone mosaic (L- and S-cones only), and a protanopic cone mosaic (M- and S-cones only). In exploring the effect of cone type, we did not reoptimize focus for the shift in spectral sensitivity between L- and M-cones, but given the high overlap in spectral sensitivity, we do not expect that this is a critical factor. Results demonstrate that, as expected, shortening integration time reduces VA in all eye models, but there is very little change in the impact of chromatic aberration correction on VA. We also found little effect of doubling macular pigment density from 0.35 to 0.70, from modeling the lens density of a 60 year old rather than a 32 year old eye, and from examining eyes with deuteranopic and protanopic cone mosaics. Note that we studied performance in the center of the fovea where stimuli near threshold were well within the tritanopic zone, such that the degradations associated with chromatic aberrations were driven primarily by L- and M-cones whose bandwidths are similar and largely overlapping (see Fig. 6).

4.2 Other factors that may influence performance

There are additional factors that we have not accounted for that may influence the impact of the correction of chromatic aberration on visual acuity. We simulate display primaries that match the experiment in [9]; however, the results could differ if other primaries were used. Also, the wavefront chosen to compute the PSF corresponded to the wavefront with a defocus value that yielded the maximum Strehl ratio, so that the retinal image had best optical quality at 555 nm, the wavelength with no chromatic aberration. Thus, we assume that the retinal image is nominally conjugate to the stimulus, i.e. that the accommodative system was properly functioning so that the retinal image was in good focus. However, it is well documented that, at least in some subjects, chromatic cues play an important role in the accommodative response [30–37]. Conversely, others have found that the accommodative response does not appear to be significantly influenced by LCA [38–40]. Still, given that correction of LCA could cause the accommodative system to malfunction, at least in some subjects and scenarios, it is plausible that the improvements in VA with LCA correction documented herein may not be realized in a pre-presbyopic eye (in an eye capable of accommodation). That is, it is possible that, in a pre-presbyopic eye, accommodative errors would negate the expected benefits of LCA correction. This possibility would need to be further evaluated experimentally.

Fixational eye movements have been shown to influence VA, in that microsaccades and fixational drift may improve VA [41,42]. Here we do not simulate eye movements. In our model, eye movements would lead to a reduction in information available to the ideal observer unless we elaborated the ideal observer to account for these eye movements. Without such elaboration, cone activation would be integrated over time, resulting in motion blur. Further work is needed to explore the impact of the moving eye on changes in VA with chromatic aberration correction. Our model also does not include photocurrent transduction, recombination of signals from cones by retinal ganglion cells or any other post-receptoral processes. Further work is needed to explore the impact of chromatic aberration correction in models that incorporate post-receptoral processes and additional sources of noise at post-receptoral stages.

Our acuity model employs a template matching paradigm, with the templates constructed as the mosaic of mean photopigment excitations. In this method, we assume that the pattern discrimination underlying the measurement of visual acuity is accomplished using a set of stored templates to determine which of the four stimulus alternatives was most likely on each trial, given the noisy cone excitations. With this ideal observer, we model a Poisson limited system where the only noise source present occurs due to the discrete nature of the photon and of cone photopigment excitations. Of course, as noted above, the human visual system is influenced by additional noise sources associated with post-receptoral processes. At the same time, a strength of the ideal observer approach is that it establishes upper limits on performance given the well-characterized limits on acuity imposed by optical blur, photoreceptor sampling, and noise in the photopigment excitations. Additional processing cannot restore the information lost at these early stages, and the ideal observer calculation provides an important benchmark against which to evaluate additional information loss incurred by post-receptoral stages.

An interesting approach to incorporating post-receptoral processes into an ideal observer analysis of the effect of optics on visual acuity was adopted by Watson & Ahumada [43]. In outline, these authors used individual observer wavefront aberration measurements to estimate the retinal image, then applied an additional blurring step to capture the overall effect of neural processing on the spatial information available for perceptual decisions. In their work, they had access to psychophysical data for subjects whose wavefront aberrations were known. To improve the correspondence between model predictions and empirical data, they adjusted the variance of Gaussian noise added as the last step in their calculation before application of an ideal observer decision rule. This approach provided a good account of the data, and thus has in common with our work the conclusion that optical factors set an important limit on acuity. Their analysis was for grayscale images and did not consider the effect of chromatic aberrations. An extension of our work would be to add a post-receptoral filtering step and to tune the noise level to fit measured data. To this end, note that the SNR of a Poisson noise limited system may be tuned by varying the integration time. In the best case, analyses of this sort would be accompanied by data from individual subjects where characterization of monochromatic aberrations, chromatic aberrations, and visual performance were all available. Another factor that might be considered in such models is the observation that human letter identification is most sensitive to information in a limited spatial frequency band [44].

4.3 Comparing simulations to experiments

The low luminance condition simulated in this manuscript was selected to match the experimental conditions in [9]. Although individual subjects’ wavefronts were not measured in that experiment, to facilitate a general comparison, we used wavefront data from eyes that represent a reasonable range of the optical quality expected in the population. With that limitation in mind, we plot the experimental and computational results together (Fig. 12). In this comparison, LCA of 2.2 D was used because the results in [9] were consistent with the population average in the literature, and vertical TCA of 1.58 arcmin was used to more closely match the results in [9] where somewhat larger than average TCA was measured for the specific subjects studied. The overarching trends are quite similar; that is, the results of the simulations are qualitatively consistent with the experimental results. Both indicate an improvement in VA when LCA is corrected. Both indicate modest, negligible, changes in VA when TCA is corrected. The small differences observed experimentally for TCA correction were not statistically significant. Both simulations and experiment indicate more of an improvement in VA when LCA is corrected than when TCA is corrected. Lastly, both indicate that the best performance is obtained when both LCA and TCA are corrected.

 

Fig. 12. Visual acuity in the prior experiment in [9] (left) and present simulations (right) with and without LCA and TCA correction. The LCA-present condition is shown in the left half and the LCA-corrected condition is shown in the right half of each plot. Each pair shows the TCA-present condition at the left and the TCA-corrected condition at the right. Simulations were conducted with 2.2 D of LCA and 1.58 arcmin of vertical TCA. The means are shown with a gray square, error bars are 90% confidence intervals for the variation within the population of subjects studied. Note that, despite having different limits, both plots span a range of 2.5-lines on the ordinate.

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Still, the match is not perfect. In the experiment, we found that correction of TCA alone improved VA by 0.02 logMAR, correction of LCA alone improved VA by 0.04 logMAR, and correction of both TCA and LCA improved VA by 0.08 logMAR. In the simulation, with 1.58 arcmin of TCA, we found that correction of TCA alone improved VA by <0.005 logMAR, correction of LCA alone improved VA by 0.09 logMAR, and correction of both TCA and LCA improved VA by 0.11 logMAR. Also, with 0.4 arcmin of TCA, the simulation gave a mean VA improvement of 0.11 logMAR with LCA correction alone. Thus, simulations differ from the experiment in the magnitude of the impact of LCA correction; namely, simulations indicate a mean improvement of 0.09 to 0.11 logMAR while experiments indicate a mean improvement of roughly half that logMAR value (0.04 logMAR). Differences that may be responsible for this disagreement in the magnitude of the impact of LCA correction include: (1) the experiment had 3 subjects whereas the simulation had 18 subjects, that is, subjects in the experiment and the simulation had different aberrations, (2) the optical quality of the experimental apparatus was not diffraction limited in all configurations, there were more aberrations present in the LCA corrected condition; these additional aberrations would be expected to degrade performance in the LCA corrected condition, thereby decreasing the enhancement with LCA correction, (3) monochromatic aberrations were assumed to be constant across the spectrum, but the results of McLellan et al. [45] are consistent with the notion that monochromatic aberrations may vary across the visible spectrum in a manner that attenuates the impact of chromatic aberrations and (4) as discussed above the ideal observer does not capture post-receptoral information loss that occurs in real subjects, nor the effect of eye movements on performance. However, prior comparisons of human and ideal observer performance often have the feature that the pattern of results is similar across the two, but ideal observer performance is systematically higher, with the difference attributed to an omnibus lower relative efficiency in the way human observers use visual information [46]. Here, the differences in improvement might be taken to indicate that the difference in efficiency is not entirely invariant across different levels of LCA and TCA, but enough potential factors are involved that we cannot draw firm conclusions about the differences. Still, we think the agreement in the pattern of the results shown in Fig. 12 suggests that ideal observer calculations provide a useful benchmark for evaluating what optical corrections are most likely to improve actual human observer performance.

4.4 Is it a paradox that ideal observer acuity can exceed the Nyquist limit?

In the high luminance condition, we found that after correcting chromatic aberrations, the subject with the best monochromatic optical quality had a VA of -0.61 logMAR (see Supplement 1). This exceptional VA exceeds what is observed clinically, which is not surprising since the ideal observer faces less information loss than human observers. Does ideal observer VA exceed the Nyquist limit imposed by spatial sampling by the cone photoreceptors, and if so, is this a paradox? In our simulations, we employ a peak foveal cone spacing of 250,000 cones/mm² which is nominally consistent with both in-vitro [23,47] and in-vivo measures [48]. With a hexagonal lattice, we have 3 cones per hexagon 250,000/3 = 8.33·10⁴ hexagons/mm², or 3/250,000 = 1.2·10⁻⁵ mm²/hexagon. The area of the hexagon, $A = 1.5\sqrt 3 {\varphi ^2}$, where φ is the center-to-center cone spacing. In this case, φ is 2.15 µm, and assuming 291 µm/degree of visual field in the fovea, this gives a center-to-center cone spacing of 0.44 arcmin. Thus, the Nyquist limited visual acuity is MAR = 0.44 arcmin = -0.36 logMAR. Given that our simulations provide an acuity of -0.61 logMAR, this indicates that the ideal observer enables visual acuity that exceeds the Nyquist limit.

We are not the first to observe that the ideal observer enables discrimination performance that exceeds the Nyquist limit. At 160 cd/m², Geisler [16] shows ideal observer two-point acuity of approximately 20 arcsec for a simulation with no blur (VA of -0.48 logMAR); for higher luminances with modest blur he shows ideal observer two-point acuity of approximately 5 arcsec (VA of -1.08 logMAR). Geisler states “from 10 to 100 cd/m², resolution is only slightly better than the diameter of a photoreceptor, but at higher luminances it becomes considerably better.” Why are these findings reasonable?

Performance better than Nyquist emerges from the classification nature of the task. Given that there are only four possible stimulus orientations, the ideal observer can correctly classify a stimulus even when its spatial structure is not resolved, as long as the mean excitations to the four stimuli differ from each other by an amount that sufficiently exceeds the noise in the cone excitations. This observation also explains why performance at low luminance is worse. Recall that the ideal observer is limited by Poisson noise, and for Poisson noise limited systems the SNR is proportional to the square root of the number, N, of photons that generate photocurrent. Thus, a larger difference in the mean excitations is required for the lower luminance stimuli, leading to lower ideal observer VA.

5. Conclusion

In conclusion, simulations indicate an improvement in VA when LCA is corrected. Improvements in VA are greater in subjects with less monochromatic aberrations and in subjects with less foveal TCA. With LCA of 2.2 D and TCA of 0.4 arcmin (nominally population average), ideal observer VA improved by -0.04 to -0.16 logMAR. Improvements in VA appear to be (mostly) linear with respect to the magnitude of LCA. The improvements in VA with correction of LCA decrease with increasing amounts of monochromatic aberrations and with increasing amounts of foveal TCA.

Simulations indicate modest changes in ideal observer VA when TCA is corrected. Improvements in VA are manifest when large amounts of TCA are present, and when LCA is fully corrected, but in the known physiological range, and in the presence of typical amounts of LCA, changes in VA with TCA correction are negligible. With vertical TCA of 2.76 arcmin (in between the population average and the population maximum) and LCA of 2.2 D, VA improved by 0 to -0.04 logMAR. Therefore, simulations indicate more of an improvement in VA when LCA is corrected than when TCA is corrected.

Finally, the modeling techniques herein offer new opportunities for thinking about optical corrections. Our simulation results exhibit the same pattern as human experimental results, demonstrating that we can use these simulation tools to formulate hypotheses about what corrections will provide maximal benefit, and to serve as a benchmark for experimental results.