Gullstrand Eye Model
This project is done by Yasin Buyukalp and Yuanyu Chen. These project partners are deeply thankful to their mentor, Professor Brian Wandell, for his guiding. They also thank to Rosemary Kim Le, Trisha Pei-Wei Lian, and Andy Lin for their helpful answers about using the main simulation tool of this project, CISET.
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
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]
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.
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).
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.
Next, we compared the linespread function of an eye in the book and longitudinal-chromatic aberration (LCA) of the Gullstrand-Le Grand eye model that simulated with the published data in Reference 9. By comparing Figure 10, with the left panel of Figure 12, it can be seen that the linespread functions are pretty comparable. Also, by comparing Figure 11 with the right panel of Figure 12, it can be seen that LCA of our Gullstrand-Le Grand eye model implementation follows a similar trend as the one in Figure 3 of Reference 9.
Later, we compared the polychromatic point spread function (PSF) of our Gullstrand-Le Grand eye model implementation as a function of aperture size with the result in Panel (a) of Figure 5 of Reference 9. By looking at Figure 13 and Figure 14, it can be seen that the right half of our figure resembles the one in Reference 9. Just like the result in Figure 13, we also observed that smaller aperture leads to a sharper PSF as it can be seen in Figure 14. However, we are surprised that the larger aperture does not allow more overall light through the lens. We are not sure why it is the case.
Finally, we compared the effects of defocus on the polychromatic modulation transfer function (MTF) of our Gullstrand-Le Grand eye model implementation with the result in Panel (a) of Figure 12 of Reference 10. The line in red is our sensor defocused, which has a similar characteristic to the defocused model result of Reference 10. Both results have successive bumps for nonzero defocus case whereas there is no bump for the zero defocus case.
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. <http://onlinelibrary.wiley.com/doi/10.1111/j.1755-3768.2011.02235.x/full>
[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). <http://www.springer.com/us/book/9783662135013>
[8] A. C. Kooijman, "Light distribution on the retina of a wide-angle theoretical eye," J. Opt. Soc. Am. 73, 1544-1550 (1983). <https://www.osapublishing.org/josa/abstract.cfm?uri=josa-73-11-1544>
[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. <https://www.osapublishing.org/josaa/abstract.cfm?uri=josaa-2-8-1273>
[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. <https://www.osapublishing.org/josaa/abstract.cfm?uri=JOSAA-16-8-1881>
[11] D. A. Atchison and G. Smith, "Continuous gradient index and shell models of the human lens," Vision Res. 35, 2529-2538 (1995). <http://www.sciencedirect.com/science/article/pii/004269899500019V>
[12] M. Herzberger, "Colour correction in optical systems and a new dispersion formula," Opt. Acta 6, 197-215 (1959). <http://www.tandfonline.com/doi/pdf/10.1080/713826287>
Appendix I
This folder includes the data file (gullstrand.dat) file for defining Gullstrand-Le Grand eye model whose parameters are taken from the Table 1 of Reference 9. You can put this .dat file to ciset-master\data\lens folder. computeGullstrandEyeRayTracing.m is our main script to perform ray tracing measurements with the model. For ray tracing results in Figure 6 and Figure 7, we used this code.
For the spherical aberration measurement in Figure 9, we performed many ray tracing simulations, and manually followed the rays coming from a particular height until the point that they focused. Then, we recorded the focal length for each ray coming from a different height. The recorded data is in sphericalAberrationMeasurement.m script. Then, by using sphericalAberrationMeasurement.m, we created the figure.
For the results in Figure 12, we used the different refractive index for all the mediums at different wavelengths. The refractive indices are found by using the Herzberger formula shown in Figure 5. At this step, we did not change the gullstrand.dat file. After loading that file, we simply changed the corresponding refractive indices inside the code of computeGullstrandEyeRayTracing.m.
For the result in Figure 14, we again performed many ray tracing measurements with different pupil size. We again changed the pupil size inside computeGullstrandEyeRayTracing.m by simply changing the parameter apertureMiddleD.
For obtaining Figure 16, we again used computeGullstrandEyeRayTracing.m by modifying it to create defocus. Then, we drew MTF corresponding to each different focal length.
Hence, computeGullstrandEyeRayTracing.m is like the master code and gullstrand.dat has the quantities of the main Gullstrand-Le Grand eye model system. For each result, we modified only computeGullstrandEyeRayTracing.m, not gullstrand.dat, such that we can make the measurement of interest.
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.