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 model this historically prominent eye model, Gullstrand-Le Grand eye model, and we perform some analysis of it by using ray tracing simulations. Then, we compare 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]
Methods
We extracted values for the Gullstrand eye model from Navarro's 1985 paper, and built a CISET lens with these values. We did ray tracing in CISET, and varied index of refraction as a function of wavelength manually in the code 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
Comparison of LSA of our Gullstrand eye model in CISET versus data from Navarro's 1985 paper. The dotted line on the right figure represents Navarro's Gullstrand-Le Grand eye model, and our values are fairly similar to his.
Comparison of linespread function and chromatic aberration of our Gullstrand eye model in CISET versus the actual human linespread function and data from Navarro's 1985 paper. The linespread functions are pretty comparable, and the chromatic aberration of our lens follows similar trends as modeled by Navarro.
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 the Gullstrand eye model in CISET correlated fairly well with the Navarro Gullstrand-Le Grand model.
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. Navarro's paper includes 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 and to get a wide-angle model
- Modeling lenses as conical instead of spherical, as is done in Navarro's 1999 paper.
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).
Appendix
File:Gullstrand.zip gullstrand.dat lens file for integrating into CISET