A formula for the mean human optical modulation transfer function as a function of pupil size

Abstract

Abstract

Abstract:

Abstract  We have constructed an analytic formula for the mean radial modulation transfer function of the best-corrected human eye as a function of pupil diameter, based on previously collected wave front aberrations from 200 eyes (Thibos, Hong, Bradley, & Cheng, 2002). This formula will be useful in modeling the early stages of human vision.

Introduction

Modeling of human vision begins with an optical component. In the past, optical effects have frequently been ignored or regarded as part of the contrast sensitivity function. In recent years there has been a growing appreciation of the specific role of optics in spatial vision and an increasing need for a simple model for visual optics. In particular, there is a need for a model of the human visual point spread function (PSF), or its Fourier transform, the optical transfer function (OTF), either of which would allow calculation of retinal images of visual displays. Because the PSF depends significantly upon the pupil diameter, the model should include this parameter. 

While actual individual optical PSFs are not symmetrical, their average is unlikely to exhibit any phase shifts, in which case the average OTF would be all real and equivalent to its modulation transfer function (MTF). Likewise, while individual MTFs are not radially symmetric, their average is likely to be approximately symmetric. In this report we develop a mathematical formula for the average human visual radial MTF as a function of pupil diameter. 

Estimates of the OTF or MTF of the human eye have employed double-pass (Campbell & Gubisch, 1966; Westheimer & Campbell, 1962), interferometric (Campbell & Green, 1965; Williams, Brainard, McMahon, & Navarro, 1994), and aberrometric methods (Thibos, Hong, Bradley, & Cheng, 2002; Walsh & Charman, 1988). Here we use the last method, making use of aberration data collected elsewhere (Thibos et al., 2002). 

A number of analytical formulas have been proposed for the MTF or PSF (Artal & Navarro, 1994; Deeley, Drasdo, & Charman, 1991; Geisler, 1984; Guirao et al., 1999; IJspeert, van den Berg, & Spekreijse, 1993; Jennings & Charman, 1997; Krueger & Moser, 1973; Navarro, Artal, & Williams, 1993; Williams, Brainard, McMahon, & Navarro, 1994). Of these, only two provide formulas for various pupils and white light (Deeley et al., 1991; IJspeert et al., 1993). The former is based largely on double-pass data from three observers. The latter formula is particularly elaborate, including effects of age and pigmentation but is based (for small angles) on the data of one observer. Our formula is an advance upon existing formulas for various pupils in white light in being based on modern measurements from a large population of eyes. 

Our approach is enabled by three recent developments. The first is the collection of wave front aberration data, for one wavelength and pupil diameter, from 200 best-corrected eyes of 100 young visually healthy observers (Thibos et al., 2002). The second development consist of mathematical techniques for computing aberrations at a smaller pupil from those collected at a larger pupil (Díaz, Fernández-Dorado, Pizarro, & Arasa, 2009; Mahajan, 2010; Schwiegerling, 2002). This allows us to extend the data of Thibos et al. (2002) to a range of pupil sizes. The third development consists of a mathematical technique for computing the polychromatic PSF from monochromatic aberrations (Ravikumar, Thibos, & Bradley, 2008). Together these three developments allow us to compute the white light MTF for a large population of individual eyes at various pupil diameters and from their average to derive an analytical formula. In conjunction with a recently proposed formula for average pupil diameter (Watson & Yellott, 2012), we may now compute the average MTF for specified viewing conditions. 

Computing the mean radial MTF

The two-dimensional complex OTF can be computed from a particular set of wave front aberrations obtained for a given wavelength and pupil diameter. Here we employ established techniques to extend this calculation to other pupil diameters and to white light. We apply these techniques to a set of 200 eyes for which aberration data were collected by Thibos et al. (2002). We compute the MTF, its radial average, and then the population radial average. We then approximate the population radial average with a mathematical formula. 

The data we used in this study consisted of wave front aberrations for 200 eyes from 100 individuals with normal healthy vision (Thibos et al., 2002). The average age of the subjects was 26.1 years. Wave fronts were measured at a wavelength of 633 nm and referenced to a 6 mm pupil. Subjectively measured sphero-cylindrical refractive errors were corrected before the wave front measurement, but subsequent measurements included residual lower order aberrations that we included in our calculations. Each set of aberrations was expressed as Zernike coefficients for the first 35 modes. 

To compute the mean radial MTF for each pupil diameter, we performed the following steps. 

1.  
   For each of the 200 eyes, starting from the Zernike coefficients at 6 mm, and using the methods described by Schwiegerling (2002) (see also Díaz et al., 2009; Mahajan, 2010), coefficients were computed for pupil diameters from 2 to 6 mm in steps of 0.5 mm.
2.  
   For each eye and pupil diameter we then computed from the Zernike coefficients the discrete OTF for polychromatic (equal energy spectrum) white light, using the methods described in Watson and Ahumada (2008, 2012) and based on methods of Ravikumar et al. (2008). Zernike radial orders of two or greater were used. We used a center wavelength of 555 nm and a wavelength spacing of 20 nm, weighted by luminance. The discrete OTF was 256 × 256 in size. The scale of the discrete OTF and PSF depends upon pupil diameter. Pupil diameters from 2 to 6 mm yield PSF image resolutions of 521 to 174 samples/deg and OTF resolutions of 2.035 to 0.678 samples/cycle/deg, respectively.
3.  
   For each OTF, we then computed the MTF as the absolute value of the OTF.
4.  
   For each MTF, we then computed the radial MTF. For each pixel in the MTF we computed its distance from the origin in cycles/deg. The complete set of pixels was then binned by distance with a bin width of 1 cycle/deg. The magnitudes within each bin were averaged. The result was a vector of magnitudes and associated spatial frequencies.
5.  
   Finally, we computed the mean radial MTF, computing the mean and standard deviation at each frequency over the 200 eyes. We considered averages of both log and linear values, but because the distributions of log values were much more skewed than linear values, we used the linear mean.

Our method of computing the radial mean from a set of OTFs (Steps 3–5) is essentially “method D” discussed by Thibos et al. (2002). The results are shown separately for each pupil diameter in Figure 1. To allow a more direct comparison of the various pupil diameters, we also plot the means together in Figure 2. 

Figure 1

 

Mean radial MTF for nine pupil diameters. The colored band indicates plus and minus one standard deviation.

Figure 2

 

Mean radial MTFs for pupil diameters from 2 to 6 mm.

Fitting formulas to the mean radial MTF

Previously, for each pupil diameter we had computed the mean and standard deviation (over eyes) of the magnitudes at each radial spatial frequency. Candidate functions were fit to the mean radial MTF for each pupil diameter by minimizing the sum over all radial frequencies of squared errors between the function and the mean at each frequency, divided by the corresponding variance at each frequency. At each pupil diameter, fitting was restricted to magnitudes greater than 0.01. This means that there may be different numbers of magnitudes at different pupil diameters. One or more parameters were estimated for each function for each pupil diameter. Finally, we computed an RMS error for the complete set of pupil diameters by summing the minimum squared errors, dividing by the total number of data points, and taking the square root. 

A number of formulas for the human average MTF have been proposed. Jennings and Charman (1997) provide a table of examples, which we extend here. These functions may be divided into two groups, which we describe as the exponential group and the Lorentzian group. Some of the formulas also make use of the diffraction-limited MTF, which we discuss first. 

Diffraction limited MTF

The MTF of an aberration-free incoherent optical system limited only by diffraction, at a wavelength λ nm and with a circular pupil with diameter d mm is given by  

 

where u is spatial frequency in cycles/deg, and where  

 

and u₀ is the incoherent cutoff frequency given by  

 

 

At a given wavelength and pupil diameter, the MTF is strictly limited to lie below the corresponding diffraction-limited MTF (MTF_DL), as defined by Equation 1. For this reason Williams et al. (1994) multiplied their candidate monochromatic formula by the MTF_DL. However the situation is more complicated for the polychromatic diffraction-limited MTF. First, the function is only truly diffraction limited at the one wavelength in focus, because chromatic aberration will necessarily intrude at the other wavelengths. Second, the diffraction limit depends on wavelength (Equation 3). We illustrate this complexity in Figure 3, which shows the MTF_DL at nine different pupil diameters for both monochromatic light at 555 nm (solid curves) and white light in focus at 555 nm (dashed curves). 

Figure 3

 

MTF_DL for nine different pupil diameters, from 2 to 6 mm in steps of 0.5 mm, for both monochromatic light at 555 nm (solid curves) and white light in focus at 555 nm (dashed curves).

For the smallest pupil of 2 mm, the monochromatic and polychromatic curves are close, while for the largest pupil of 6 mm, they are far apart. This illustrates the increasing effect of chromatic aberration as the pupil enlarges. However, the two curves always converge at the monochromatic limit. For this reason we have included the monochromatic 555 nm MTF_DL as a component in our fitting to effectively restrict all functions to the region below this monochromatic DL limit. However, we found that multiplying by the monochromatic MTF_DL was not as effective as the square root of that function. Accordingly, for each function, we have considered the fit of the function by itself, and when multiplied by MTF_DL, and by √MTF_DL. 

Exponential family

Comprising Functions 1–5 in Table 1, the exponential family includes (1) simple exponentials (Fry, 1970), (2) Gaussians (Fry, 1970; Geisler, 1984; Krueger & Moser, 1973), (3) what have been called generalized Gaussians with an arbitrary exponent γ (Deeley et al., 1991; Jennings & Charman, 1997; Johnson, 1970; Rovamo, Kukkonen, & Mustonen, 1998; Watson & Ahumada, 2005), (4) sum of an exponential plus a constant (Williams et al., 1994), and (5) the sum of two exponentials (Artal & Navarro, 1994; Navarro et al., 1993). 

Table 1

 

Candidate formulas shown with number of parameters n per pupil diameter and total RMS error. Error is shown for the formula alone and when multiplied by MTF_DL or by √MTF_DL. The symbol u is spatial frequency in cycles/deg, and u₁, u₂, α, β, and γ are parameters.

	f(u)	n	RMS
				MTFDL	√MTFDL
1	exp[−u/u1]	1	1.56	1.24	1.25
2	exp[−(u/u1)2]	1	1.94	1.74	1.82
3	exp[−(u/u1)γ]	2	0.79	0.98	0.86
4	α exp(−u/u1) + (1 – α)	2	1.25	0.65	0.73
5	α exp[−u/u1] + (1 – α)exp[u/u2]	3	1.21	0.69	0.71
6	[1 + (u/u1)2]−3/2	1	1.22	1.30	1.22
7	[1 + (u/u1)2]−1	1	0.86	1.00	0.80
8	[1 + (u/u1)2]−0.62	1	1.21	0.62	0.37
9	[1 + (u/u1)2]−β	2	0.84	0.44	0.33
10	[1 + (u/u1)2]−β(MTFDL)γ	3	0.16

Lorentzian family

Comprising Functions 6–10 in Table 1, the Lorentzian family includes (7) the Lorentzian, which is the Fourier transform of an exponential line-spread function and also the distribution function of the Cauchy probability distribution, (6) a version in which the exponent is −3/2 and which is the Hankel transform of the exponential, (9) a generalized Lorentzian in which the exponent −β is arbitrary, and (8) a version in which the exponent is fixed at the value of 0.62. In these Lorentzian family functions, it is worth noting that at high frequencies (u ≫ u₁) the value is proportional to u^−2β, which is a straight line on log-log coordinates with a slope of −2β. The final entry (10) is a version of the Lorentzian in which the scale parameter u₁, the exponent β, and the exponent γ on the MTF_DL are all free to vary. 

Results

In Table 1 we show for each formula the number of parameters (per pupil diameter), followed by RMS error for the formula alone, when the formula is multiplied by MTF_DL, and when multiplied by √MTF_DL. The smallest RMS error is for the version (Formula 10) in which there are three parameters. However we sought a formula with fewer parameters, in part because when there are many parameters, they tend not to be well-behaved with respect to pupil diameter. As will be seen below, we would also like to model the parameter variation with pupil diameter. 

None of the prior formulas (1–5) provided good fits, especially when number of parameters is considered. In most cases, the inclusion of the MTF_DL or √MTF_DL term reduces the error, sometimes by a large amount. Of the formulas we considered with two or fewer parameters, the lowest error (RMS = 0.33) was for Formula 9, using √MTF_DL. The error was almost as low (RMS = 0.37) for a variant (Formula 8, multiplied by √MTF_DL) in which the second parameter (the exponent on the Lorentzian) is fixed at 0.62. 

General formula

Given its low error and small number of parameters we have adopted Formula 8 (with √MTF_DL) as the starting point for our radial MTF formula. We plot the estimated parameter u₁ for Formula 8 for the nine pupil diameters, as shown in Figure 4. This set of points is fit reasonably well by a second order polynomial,  

 

as shown by the black curve.  

Figure 4

 

The parameter u₁ estimated for Formula 9 at each of nine pupil diameters (red). Also shown is the best fitting second order polynomial (black).

Using Equation 4, we can construct a general formula for the radial MTF as follows  

 

 

The predictions of the general formula are shown along with the mean radial MTFs for each pupil diameter in Figure 5. These fits are reasonably good, especially considering the variability over observers pictured in Figure 1. 

Figure 5

 

Fit of the radial MTF formula (red) to the mean radial MTFs for nine pupil diameters (black points). For each diameter, we show linear and log plots. The dashed gray line shows the diffraction-limited polychromatic MTF.

Discussion

Comparison with prior formulas

As noted above, two formulas have previously been developed for the mean human polychromatic (white light) optical MTF for as a function of pupil diameter. In Figure 6 we compare those two formulas with ours, at pupil diameters of 2 and 6 mm. In the case of IJspeert et al. (1993), we have assumed an age of 26.1 years, equal to the mean age of Thibos et al.'s (2002) subjects, and a “pigmentation parameter” of 0.142, appropriate for the Caucasian population mean. 

Figure 6

 

Comparison of three formulas at two pupil diameters.

Our formula differs significantly from both previous formulas. The greatest differences are with IJspeert et al. (1993) at the smallest pupil size and Deeley et al. (1991) at the largest pupil size. Part of the difference with IJspeert et al. is that they modeled scattered light, which will attenuate lower frequencies (see Scatter below), but this only explains the difference at the lowest spatial frequencies. We do not have an explanation for the large discrepancies at the highest frequencies. Their formula was based (for small angles or higher spatial frequencies) on the data of a single observer of Campbell and Green (1965), using an interferometric method. Williams et al. (1994) have discussed possible differences among methods of estimating the optical MTF, noting that interferometric MTF is generally above the double-pass MTF. 

Residual astigmatism

Prior to their wave front measurements, Thibos et al. (2002) conducted a subjective refraction and inserted appropriate sphero-cylindrical correction. Despite this procedure, their wave front measurements revealed some residual low order aberrations. We have included these aberrations in our calculations, because they presumably reflect the typical low order aberrations present in even well-corrected observers. They noted that the distribution of residual astigmatism had a slight bias towards positive J₀, or “with the rule.” In Figure 7 we plot the log relative MTF gain as a function of orientation. This was obtained by computing the mean discrete MTF, as described above, and then collecting all values in bins of width 1 cycle/deg in radial frequency, restricting to values below 50 cycles/deg. The values within each collection were then normalized by their mean and plotted against orientation. The results for the 50 collections are combined in Figure 7. The result shows a relative increase in sensitivity, by almost a tenth of a log unit (almost 25%), at an orientation of 90°. 

Figure 7

 

Variation in mean MTF gain as a function of orientation.

Scatter

The OTF (or PSF) derived from wave front aberrations is not a complete description of the optical transfer properties of the eye. To it must be added the effects of light scatter (Ginis, Pérez, Bueno, & Artal, 2012; Piñero, Ortiz, & Alio, 2010; van den Berg, Franssen, Kruijt, & Coppens, 2012). From a practical point of view, the effect of scatter is to reduce the contrast of retinal images. In discussions of scattered light and visual glare it is common to distinguish between small-angle (< 1°) and large-angle components of the PSF. Scatter, which is the typical source of the large-angle components, may contribute very small values to the PSF, but when integrated these components may constitute as much as 16% of the PSF volume in the normal young eye, and even more in older eyes (Ginis et al., 2012; IJspeert et al., 1993). This means that the MTF derived from aberrations, which by definition asymptotes at one for low spatial frequencies, overstates the actual contrast of retinal images. 

The fraction of incoming light that is scattered (the fraction of the PSF volume that is due to the large angle components) is typically specified by the “straylight parameter” S. The fraction that is not scattered (fraction of the PSF volume that is due to the small angle components) is 1 − S. Thus in calculations of retinal contrast that include effects of scattered light, the MTF provided by our formula should be multiplied by the factor 1 – S. From IJspeert et al. (1993) we can derive an expression for this factor,  

 

where y is age in years, y₀ = 70 years, and p is a parameter related to skin and iris pigmentation (and equal to the fraction of light scattered at age zero). The log of this expression is plotted in Figure 8 for the two extreme values of the pigment parameter quoted by IJspeert et al. for the “mean blue caucasian eye” (p = 0.16) and the “mean pigmented-skin dark-brown eye” (p = 0.056). For the former case, scatter reduces contrast by between about 0.08 and 0.3 log units, depending on age.  

Figure 8

 

Contrast attenuation as a function of age for two pigment factors.

Simulating retinal images

A goal of this project was to provide a means to simulate filtering by the eyes optics as a function of pupil diameter. Because the MTF provides only the magnitude term of the OTF, and because we are computing a radially symmetric MTF, we must assume that the phase is zero. In that case, retinal images can be simulated by multiplying the two-dimensional Fourier transform of the object image by the model MTF (expanded to two dimensions by rotation), and inverse transforming. As an example in Figure 9 we show a letter at the acuity limit filtered by the model MTF with various pupil diameters. 

Figure 9

 

Letter Z from the Sloan font with a size of 1/12° (logMAR = 0) filtered by the model MTF at a pupil diameters of 2–6 mm.

Average performance versus performance of the average

One concern regarding the use of an average MTF is that it may not accurately simulate the average performance of a population of observers. Elsewhere we have advocated instead averaging the performance of a population of simulated observers as a way of simulating average observer performance. In the present case, this concern is particularly strong because while each individual observer has phase distortions, unique to their own wave front aberrations, the average MTF has none. Also, as Thibos et al. (2002) have pointed out, there is no one unique best way of averaging OTFs. 

Because of these concerns we compared the average acuity performance of a population of simulated observers to the performance of a single simulated average observer. The average observer was either computed from the formula or was the numerical mean MTF. 

To estimate acuity from a given OTF, we generated a set of Sloan letter images varying in size in half octave steps. We filtered the letters by the OTF and then simulated performance at each letter size using our template model of letter identification (Watson & Ahumada, 2008, 2012). We used a fixed noise power spectral density (PSD) of 12 dBB (Watson & Ahumada, 2012) to approximate average visual acuity. From the psychometric function relating size to percent correct, we estimated acuity as the size yielding 75% correct. We express acuity in units of letters/deg (the inverse of the letter size in degrees). 

We applied this method in three cases. In the first case we estimated acuity for each of the 200 OTFs and then computed the mean acuity at each of five pupil diameters (mean acuity). We then estimated acuity for the single empirical mean MTF obtained from the 200 OTFs, for each pupil diameter (mean MTF). Finally we estimated acuity from the single MTF computed from the formula at each pupil diameter (formula). 

The results are shown in Figure 10. As expected, simulated acuity declines as pupil diameter increases for all three cases. For all diameters, the three results are essentially the same. These results are somewhat surprising. They suggest that phase shifts, present in individual eyes but absent in the average, are not very significant in the acuity task. This may be because in our model of acuity the templates include whatever phase errors the wave front aberrations of the observer introduce. But in summary, the results reassure that the formula for the mean MTF produces results close to that of the average acuity performance of the population. 

Figure 10

 

Acuity at five pupil diameters simulated by three methods. Error bars show ±1 standard deviation about the mean acuity results.

It is important to note that the simulations in Figure 10 only serve to demonstrate the similarity between the mean performance and performance of the mean. The variation in acuity with pupil diameter considers only variations in optical imaging and does not take into account possible changes in the effective PSD due to changes in retinal illuminance or visual adaptation. The threefold change in diameter will result in a nine-fold change in retinal illuminance if luminance remains constant. 

Limitations

With respect to predictions of visual performance outside of the laboratory, our polychromatic, luminance-weighted, MTF formula is an advance upon monochromatic formulas. However, strictly speaking, it is appropriate only for chromatically homogeneous images with an equal energy “white” spectrum. While this is approximately true for the many achromatic stimuli used in research on spatial vision, it is less valid for a colorful real world in which the eye may accommodate differently for focal targets of different colors (Autrusseau, Thibos, & Shevell, 2011). Nevertheless, our MTF may serve as an initial approximation or mean in such circumstances. 

Demonstration

To allow the reader to visualize changes in the MTF due to variations in pupil diameter, and to allow comparisons with other formulas, we provide an interactive demonstration in Figure 11. 

Figure 11

 

Interactive demonstration of the formula for mean human optical MTF.

Conclusions

Using monochromatic wave front aberration data for 200 eyes, and mathematical techniques for extrapolation to polychromatic light and other pupil diameters, we have produced mean radial white light optical MTFs for eight pupil diameters between 2 and 6 mm. 

The mean radial MTF at each pupil diameter has been fit with a variety of analytic functions. A good overall fit is provided by a simple one-parameter function. That parameter was found to be a quadratic function of pupil diameter. Together these results yield a simple analytic formula for the mean radial optical MTF of the best-corrected human eye. 

Supplementary Materials

xlsx S1 

txt S2 

png S3 

xlsx S4 

Demonstration of a formula for the Human Optical MTF 

This page links to a demonstration for use with the article: Watson, A. B. (2013) . A formula for the mean human optical modulation transfer function as a function of pupil size. Journal of Vision 

The demonstration requires the use of the free CDF Player browser plug-in. Please download the player and then reload this page. The demo file can also be downloaded and run as a standalone appllication, provided that the CDF Player is installed. 

Questions should be addressed to [e-mail]. 

Acknowledgments

I thank Larry Thibos, Albert Ahumada, Jeffrey Mulligan, and Floris van Nes for comments on the manuscript. This work supported by the NASA Space Human Factors Research Project WBS 466199. 

Commercial relationships: none. 

Corresponding author: Andrew B. Watson. 

Email: [e-mail]. 

Address: NASA Ames Research Center, Moffett Field, CA, USA. 

Appendix

Larry Thibos has kindly allowed me to include the wavefront aberration data from the Indiana Aberrations Study (Thibos et al., 2002) as a supplement to this paper. They are provided as an excel file with internal documentation by Dr. Thibos.