Gullstrand Eye Model

Introduction

Allvar Gullstrand was a Swedish ophthalmologist.[1] He received the Nobel Prize in Physiology or Medicine in 1911 for his work on the dioptrics of the eye.[2] So far, he is the only ophthalmologist who has been given a Nobel Prize for work in ophthalmology.[3] Although his most important work was perhaps finding the intracapsular accommodation as he gave his Nobel Lecture about it [4], he is also well remembered by his famous eye model [5] that carries his name.

In many fields such as physiological optics, optical design, digital image processing, and robotics, the behavior of the optical system of an eye as a part of the whole system must be taken into consideration. Throughout the history, many different eye models including Helmholtz [6], Gullstrand [5], Gullstrand-Le Grand [7], Kooijman [8], Navarro [9-10], and many more have been developed. Some of the models take into account the aspheric nature of eye surfaces as well as the shell structure of the lens to have a good agreement with measured values of eye aberrations, but they suffer from increasing complexity and involvement of large number of parameters. Thus, there is a great interest to model the eye by a simpler model for first order calculations. As a simpler yet prominent eye model, Gullstrand eye model has been widely used for first-order calculations after being revised by Le Grand. It uses a schematic two-surface lens to model the optics of the eye. In this model, the surfaces are spherical and an the mediums of eye are described by an effective index. Although this model does not agree with the measured values of eye aberrations as much as the more complex eye models, it still has a very good agreement for calculating on-axis performance. Also, it predicts the refractive power of the lens well.[9]

Hence, in this project, as our first exposure to geometrical eye models, we modeled this historically prominent eye model, Gullstrand-Le Grand eye model, and we performed some analysis of it by using ray tracing simulations. Then, we compared our results with the published data based on this model to verify our findings.

Background

Figure 1. Schematic horizontal section of the eye

Figure 2. Schematic horizontal section of the Gullstrand eye model

It is well known that an eye has a gradient-index (GRIN) crystalline lens, which is sometimes approximated by a shell structure.[11] This can be noticed by looking at the horizontal section of a realistic eye such as the one shown in Figure 1. Here, we see that human lens does not have a uniform refractive index. Nevertheless, the exact distribution of the refractive index of the human lens is not well known yet.[10] Also, taking the shell structure of a human lens into account in an optical eye model increases the complexity of a model to describe and simulate that model.[9] These models have several adjustable parameters, which are not ideal to perform a first-order fast calculations to estimate the basic optical properties of the eye.

Ideally, an eye should reproduce both anatomical and optical properties accurately. [10] However, such a model is highly complicated. Also, there is not enough experimental data to decide important parameters to build such a model.[10] As always, in these kind of modeling problems, often there is a trade-off between accuracy and simplicity. In this respect, although Gullstrand eye does not incorporate shell structure of a human lens, it has been a historically prominent model for its simplicity while allowing to compute acceptable first order results. It uses an effective refractive index to describe a human lens, and hence it exhibits higher aberrations than normal emmetropic eyes.[10]

A Gullstrand eye model uses a two surface-lens to describe the optical operation of the eye. The schematic of this eye model can be seen in Figure 2. Here, there is an aperture in the middle, which represents the pupil. There are two spherical surfaces in front of the pupil, and another two spherical surfaces behind the pupil. Each surface is described by its radius of curvature. Each medium separated by these surfaces are described by its index of refraction. The model also describes the distance between the centers of the successive surfaces.

The first surface represents the anterior surface of the cornea whereas the second represents the posterior surface of cornea. Similarly, the third surface represents the anterior surface of lens whereas the fourth surface represents the posterior surface of the lens. Inside the region which is in front of the first surface, there is the scene that the eye is looking for. The region between the first two surfaces represents the cornea. There is aqueous humour between the second and the third surface. The lens is represented by the region between the third and the fourth surface. After the fourth surface, there is vitreous humour until the retina. In this model, retina is not taken into account. Only thickness of the vitreous humour is considered. So, the model is interested in how the image of the scene would be at the end of the vitreous humour given the parameters. Also, the model allows to analyze some optical properties of the eye system such as spherical aberrations. Moreover, refractive indices of the materials are taken differently for each wavelength, which gives a first-order estimation opportunity for calculating chromatic aberrations.

The necessary parameters to describe the Gullstrand-Le Grand eye model is taken from Table 1 and Table 2 of the Reference 9.[9] For the sake of completeness, these tables are shown in Figure 3 and Figure 4.

Table 1 also has the parameters for two other eye models, Navarro and Kooijman models, which also use this two surface-lens schematic eye model. Especially Navarro's model has better match with experimentally measuredoptical properties of the eye comparing with the other two models, yet it is more complicated model since it uses surfaces with asphericity. Moreover, it is basically an updated two surface-lens eye model, which is historically based on the Gullstrand eye model. So, for these reasons, we concentrate on the implementation of Gullstrand-Le Grand eye model in this project.

Table 2 has the refractive index values of the mediums of interest in four different wavelengths. By using these four values of a medium, the refractive index values of that medium for any other wavelength is found by the Herzberger formula. [12] The expression of this formula is again taken from the Reference 9, and shown in Figure 5 below.[9]

Figure 3. Table 1 of Reference 9

Figure 4. Table 2 of Reference 9

Figure 5. Herzberger formula [9]

Methods

For our simulations, we used The Computational Image Systems Engineering Toolbox (CISET) and The Image Systems Engineering Toolbox (ISET) packages. Specifically, we built the Gullstrand-Le Grand eye model system by using CISET based on the parameters in Figure 3 and Figure 4, and the formula in Figure 5. The .dat file that builds this eye model system in CISET is attached.

Then, we performed ray tracing simulations by using CISET. During these simulations, for visualization purposes, we use some of the functions inside ISET.

While performing these simulations, we varied index of refraction as a function of wavelength manually inside our script since the lens data file can only accommodate the aberration-free model. The data from ray tracing was ported to ISET, from which we could plot the resulting image and do analysis on the sensor image. Using this method, we measured various characteristics of the eye such as the spherical aberration, chromatic aberration, and linespread function.

Results

Our first result is the ray tracing path of the Gullstand-Le Grand eye model that uses the parameters in Figure 3. As can be seen, there is a clear focusing very close to the retinal position. This is maybe a small, but still a signature of the fact that the model works.

Figure 6. Ray tracing of the Gullstrand-Le Grand eye model from the first surface until the last surface. The model uses the parameters in Figure 3

Figure 7. Ray tracing of the Gullstrand-Le Grand eye model starting from just behind the last surface until the sensor film in the retinal position. The model uses the parameters in Figure 3

Next, we measured longitudinal-spherical aberrations of our eye model. Figure 9 shows what are spherical aberrations. Basically, paraxial rays and marginal rays coming to a lens do not focus on the same spot. The lateral difference between their focus points define longitudinal-spherical aberrations (LSA).

Figure 8. Spherical aberrations

Comparison of longitudinal-spherical aberrations (LSA) of our Gullstrand-Le Grand eye model versus the LSA result of the same model published in Reference 9. Among all the lines in the right panel of Figure 8, the dotted line represents the Gullstrand-Le Grand eye model LSA result inside the Reference 9. As cen be seen, the shape of two curves are fairly similar. However, in our result, we plotted Wavelength vs Height curve whereas Reference 9 plotted LSA in diopters vs Height curve.

Figure 9. Longitudinal-spherical aberrations (LSA) of the Gullstrand-Le Grand eye model using the parameters in Figure 3. The figure in the left panel represents our result whereas the figure in the right panel represents the result published in Reference 9.

Next, we compared the linespread function and longitudinal-chromatic aberration (LCA) of the Gullstrand-Le Grand eye model that simulated with the published data in Reference 9. The linespread functions are pretty comparable, and LCA of Gullstrand-Le Grand eye model implementation follows a similar trend as the one in Figure 3 of Reference 9.

Figure 10. Linespread function of the Gullstrand-Le Grand eye model

Figure 11. Longitudinal-chromatic aberrations of the Gullstrand-Le Grand eye model

Figure 12. Longitudinal-chromatic aberrations of the Gullstrand-Le Grand eye model

Comparison of polychromatic PSF of our Gullstrand eye model as a function of aperture in CISET versus data from Navarro's 1985 paper. We also see that smaller aperture leads to a sharper PSF. However, we're not sure why the larger aperture does not allow more overall light through the lens in either model.

Comparison of the effects of defocus on the polychromatic MTF of our Gullstrand eye model in CISET versus data from Navarro's 1999 paper. The line in red is our sensor defocused, which has a similar characteristic to Navarro's defocused model.

Conclusions

We tested a Gullstrand eye model with on-axis ray tracing measurements on a planar sensor. Based on the results we measured, we concluded that our results of Gullstrand-Le Grand eye model correlated fairly well with results of the Gullstrand-Le Grand eye model published in Reference 9.

In general, we learned how Gullstrand eye model works, and how to perform ray tracing simulations. This was our first exposure to geometrical models of eye. We also learned what the relevant measured quantities in this field, and how to measure them. We learned lots of background information about eye and its anatomy, which were very different than our majors. Thus, it was a very beneficial project for us.

We did not quite know how to transform from focal length measurements into the diopters. This is because our system has composed of two lens. Also, the refractive indices at each medium is different. So, traditional diopter equation for a single lens with the same medium at both sides do not work in our case. We were unsure about converting our results into diopters correctly, so we did not do it.

We think that the explanation about how to use ray tracing in CISET should be updated. We found that there were some confusing statements in Matlab comments. That's why we had to make many trials to figure out setting up the correct system.

Some ideas for future work include:

• More modeling of field angle - our models were typically on axis measurements. We did a few off-axis measurements, but did not extract useful data out of them.
• Accommodation dependent measurements - our model does not take into account accommodation, which a real eye does. Reference 9 and 10 include equations for accommodation, and the eye model can thus be extended to account for these equations.
• Use of a spherical sensor surface instead of a planar sensor surface in ray tracing to model the retina better and to get a wide-angle model.
• Modeling lenses as conical instead of spherical, as it is done in Reference 10.

References

[1] "Gullstrand, Allvar." Complete Dictionary of Scientific Biography. 2008. Encyclopedia.com. (December 11, 2015). <http://www.encyclopedia.com/doc/1G2-2830901777.html>

[2] "The Nobel Prize in Physiology or Medicine 1911". Nobelprize.org. Nobel Media AB 2014. Web. 12 Dec 2015. <http://www.nobelprize.org/nobel_prizes/medicine/laureates/1911/>

[3] Ehinger, Berndt, and Andrzej Grzybowski. “Allvar Gullstrand (1862–1930) – the Gentleman with the Lamp.” Acta Ophthalmologica 89, no. 8 (December 1, 2011): 701–8. doi:10.1111/j.1755-3768.2011.02235.x.

[4] "Allvar Gullstrand - Nobel Lecture: How I Found the Mechanism of Intracapsular Accomodation". Nobelprize.org. Nobel Media AB 2014. Web. 12 Dec 2015. <http://www.nobelprize.org/nobel_prizes/medicine/laureates/1911/gullstrand-lecture.html>

[5] A. Gullstrand, appendix in H. von Helmholtz, Physiologische Optik, 3rd ed. (Voss, Hmaburg, 1909), Bd. 1, p. 299.

[6] H. von Helmholtz, Physiologische Optik, 3rd ed. (Voss, Hmaburg, 1909).

[7] Y. Le Grand and S. G. El Hage, Physiological Optics (Springer-Verlag, Berlin, 1980).

[8] A. C. Kooijman, "Light distribution on the retina of a wide-angle theoretical eye," J. Opt. Soc. Am. 73, 1544-1550 (1983).

[9] R. Navarro, J. Santamaría, and J. Bescós, “Accommodation-dependent model of the human eye with aspherics,” J. Opt. Soc. Am. A, vol. 2, no. 8, p. 1273, Aug. 1985.

[10] I. Escudero-Sanz and R. Navarro, “Off-axis aberrations of a wide-angle schematic eye model,” J. Opt. Soc. Am. A, vol. 16, no. 8, p. 1881, Aug. 1999.

[11] D. A. Atchison and G. Smith, "Continuous gradient index and shell models of the human lens," Vision Res. 35, 2529-2538 (1995).

[12] M. Herzberger, "Colour correction in optical systems and a new dispersion formula," Opt. Acta 6, 197-215 (1959).

Appendix I

File:Gullstrand.zip gullstrand.dat lens file for integrating into CISET

Appendix II

Both group members searched the literature together to learn about Gullstrand eye model. Then, they both independently formed this eye model system in CISET and verified by comparing each others' model. They both conducted ray tracing simulations. Then, they discussed and decided together what results to obtain for the project, and shared these final result obtaining workload as well as preparing presentation and writing report workloads in a balanced way. They feel that it is hard and perhaps even not fair for them to give the credibility of any result or work to one of them since at each step both of them have discussed and planned together. At the end, the partners see this project as implemented by together in a balanced way.