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Abstract - The cornea alone contributes to approximately 75% of the total refraction of the eye. Nevertheless, there is very few knowledge of its contribution to the optical aberrations of the images formed at the retina. Most corneal diagnostic equipments (Corneal Topographers and Keratometers) measure only its dioptric power (proportional to curvature) and shape, but do not calculate or estimate the corneal contribution to the quality of image formation at the retina. On the other hand, auto-refractors and wave-front devices measure the optical aberrations of the eye as a single optical “device”, and therefore do not indicate individual contributions. We have developed algorithms that read elevation data from a commercial corneal topographer and use this data to construct a 3D model of the eye using sophisticated computer graphics techniques. A ray-tracing procedure using this 3D model allows us to estimate the image quality formed at the retina.

Index Terms — eye model, visual system simulation.

I. INTRODUCTION The interest for the physiological and optical properties of the

human eye, and how they are related in terms of visual quality, comes from ancient times. A thorough study was undertaken by Helmholtz in the 19th century and was compiled by himself in the famous collection Helmholtz’ Treatise in Physiological Optics [1]. From this time up to modern days an incredible amount of techniques and instrumentation for visual quality measurements were implemented. Today they form a collection of tools that aid the eye care professional in providing the best diagnostic and treatment to their patients.

With the advent of refractive surgeries for correction of myopia in the early 80’s, better instrumentation was required in order to analyze the pre and post shape of the entire corneal surface. The earlier techniques such as manual keratometers [2, 3] which measure only the central 3 mm, were no longer sufficient. With the advent of more powerful and price attractive microcomputers a new line of equipments for corneal surface analysis started to take place of the conventional keratometers. These instruments, popularly known as Corneal Topographers, are based on the 19th century Placido Disc [4], but surface curvature and calculations are based on sophisticated image processing [5, 6] and computer graphics techniques [7]. They allow the eye care professional to analyze a 8-10 mm in diameter region over the cornea, displaying curvature data for thousands of points.

The first refractive surgery techniques (RK – Radial Keratectomy) were based on the application of radial incisions to flatten the central cornea. This was an empiric method and based on data collected from data gathered from in mortum eyes and animal research. During this same period certain companies and research laboratories started to investigate an alternative method for corneal intervention. At the end of the 1980’s and beginning of the 1990’s some companies made available for tests the first excited dimmer laser for corneal tissue ablations. These lasers became popularly known as excimer lasers and the first generations were primarily designed for myopic correction, and procedures became commonly known as PRK – Photo-refractive

Keratectomy. These lasers rapidly took over the place of conventional RK techniques, which were very aggressive to the corneal tissue since incisions could go as deep as 90% of the corneal depth. In the late 1990s and beginning of this century a series of technological advances allowed refractive surgery to be taken to higher levels of precision and therefore patient satisfaction. Excimer lasers with flying spot beams and eye tracking systems (to eliminate misalignment of laser treatment caused by involuntary eye movements) provide sufficient precision for practically sculpting the cornea to any desired shape [8] and a high resolution auto-refractor inspired in astronomical instruments [9] measures the eye’s wave-front optical aberrations with such a high precision, that refractive surgery can now be undertaken in a patient-to-patient base, allowing what became known as customized ablations.

With the advent of these techniques other important questions arrived. For instance: what is the best correction for the aberrations of an eye? Does the lens accommodation process influence on this choice? If it does, than should one determine the best ablation profile based on age factors, such as presbyopia? These questions still have no answer and in this work we’re attempting to solve some of them by applying ray-tracing techniques to a mixture of mathematical/biological eye model. As we will see, our preliminary eye model, different from those in literature, has in vivo corneal topography data applied to a schematic eye globe, based on statistical data. We believe this technique can lead to further contributions to the understanding and planning process for customized refractive surgeries.

II. MATERIAL AND METHODS In this work we used an instrument called the Corneal

Topographer (or Videokeratographer) to collect in vivo data from the cornea of a voluntary patient. This instrument makes possible the understanding of more complex imperfections of the cornea directly related to subtle differences in the curvature of the surface of the cornea, but does not inform us how the corneal imperfection degrade the retinal image. Figure 1 shows the launching of Placido rings in eye using the Corneal Topographer.

Figure 1. Placido rings in cornea.

A 3D simulation of the optical properties of the human eye using in vivo anatomical

data of the cornea

R. S. Duran, L. A. Carvalho, L. G. Nonato, F. A. Margarido, O. B. Martinez

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Using topographic data from the in vivo corneas of voluntary patients with different pathologies, we were able to construct a three-dimensional model of each surface (see Figures 4-7). Afterwards we simulated the ray-tracing process to quantify how the imperfections on the corneal surface are propagated to the retinal image. The data are constituted of two type text archives, contend the polar coordinates of each point showed by the topographer. The modeling of the cornea follows a project of linear interpolation, triangulating directly the points contained in the data. Figure 2 shows the general scheme of triangulation, taking every point in a degree and triangulating with the points in the next degree. Figure 3 shows the computational model of a cornea and the normal vectors computed in the vertices of the mesh. Figures 4-7 exemplify different corneal models with various types of irregularities and different levels of resolution. This variation in resolution was necessary because the linear approximation produced long triangles in the model with high resolution, generating numerical errors in intersection algorithms.

Figure 2. General Scheme of triangulation.

Figure 3. Three-dimensional Model of the cornea and the normal vectors computed.

Figure 4. Cornea model with Ceratocone in low resolution.

Figure 5. Cornea model with strong astigmatism in low resolution.

Figure 6. Cornea model of a normal human eye in low resolution.

Figure 7. Cornea model with strong astigmatism in high resolution.

In order to construct a complete model of the human eye, a model of the retina based a schematic eye proposed by Emsley [11] was constructed using several layers with successive refinement. Emsley’s schematic eye was adopted essentially because of its simplicity. The eye is considered as a single refracting surface with refraction power of 60 D and a cornea with

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refraction index of 1.33. Figure 7 is a diagram showing the parameters of Emsley’s schematic eye.

Figure 8. Emsley Reduced Eye Model

Such segmentation is used in way to simulate the distribution of the photoreceptors in the retina. In this way different eye surfaces and regions may have different refinements, such that the concentration of vertices in the cornea and the fovea can be greater than that of the peripheral retina. And the user may set up all these initial parameters. The initial step consists of determining 4 points in cross section where the cornea will be attached and a single point at the end of the model, as show in Figure 9. According to user parameters of how many layers the model will have, new reference planes are inserted as show in figures 10 and 11. The process of refinement takes each triangle in mesh and subdivides it in 4 new ones. A parameter set by user tells how many refinements the most refined layer, at fovea, will have. The previous layers will have one refinement level less, and so on. Figure 12 shows a complete retinal model with 4 layers e 5 refinements at fovea layer and figure 13 shows the complete eye model with cornea and retina.

Figure 9. Initial state of refinement process.

Figure 10. Retinal model with two layers.

Figure 11. Retinal model with three layers.

Figure 12. Final retinal model with four layers and complete refinement process.

Figure 13. Model of the human eye globe.

This method of construction of the computational model made it flexible in terms of simulation possibilities. This flexibility comes from the fact that it is possible to easily substitute the parameters for the corneal and retinal models, making it possible to easily load different corneas from different Videokeratography files. This allows the user to easily compare the retinal image for

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corneas with different aberrations (keratoconus, astigmatism, etc) and also how schematic eye parameters affect the results [11, 13, 14]. The simulation possibilities increase when we consider the techniques for ray launching, given that these rays may be arbitrarily configured, differently from previous works [12, 17], increasing the simulation possibilities. All the modeling process was made using C++. For the manipulation of the triangular mesh we used the structure named Singular Handle-Edge [16], which is a topological structure of data that facilitates the searching operations for neighborhood information. The model was stored in archive using the VTK (Visualization Toolkit) Unstructured Grid data type (http://public.kitware.com/VTK/).

III. RESULTS The simulations were implemented using the computational

model presented previously. In order to analyze how the corneal irregularities affect the formation of the retinal image, a process of ray-tracing was implemented.

The ray-tracing process uses the algorithm of intersection of polygons introduced by Badoel [10]. This algorithm consists of two stages: first it tests if rays intercept the polygon; the second stage consists of determining the coordinates of the intersection in case the first stage has had positive reply. After the process of determination of the point of intersection on the cornea, Snell’s Law is used to calculate the angle of refraction and the new ray that will be projected on the retina. Figure 14 illustrates the process of launching of a beam of parallel rays at the cornea from an arrow object, aiming such as to simulate the launching of paraxial rays. At this same figure we may see the formation of the retinal image of the blue arrow, and we may notice that the arrow has its direction inverted, as should happen.

Figure 14. Ray-tracing process.

Figure 15. A magnified view of the retina.

As we may see at Figure 15 a region of the retina is magnified. The red patterns correspond to the triangular meshes of the retina model. As may be noticed, our retinal model has very high resolution and it allows us to analyze with precision the points of intersection, and therefore will in the near future also enable us to extract quantitative information from our data.

IV. DISCUSSION It is important to notice that the simulations implemented in

this work are not actual measurements of the retinal image quality of in vivo patients, but otherwise a close study of the aberrations that may be present in common eyes found in the healthy population. This is so because we used in vivo corneal topography data in a schematic eye that has dimensions based on mean statistical data of the population. Another important fact is that we do not consider here the lens and therefore accommodation factors, and also differences in refractive indexes of the interior components of the eye. An equivalent refractive index is used for the entire eye in the accommodated state, which we know is not the case for the human in vivo eye. Nevertheless this model is a reasonable approximation to the human accommodated eye.

More accurate eye models [18] and even Physical optics (using waves) [19] should be considered, taking into account not only the shape of the anterior and posterior lens surface and thickness, but also the variation of these parameters with accommodation [20] and with age [21, 22]. Also, by collecting results from a wave-front device and the corneal topographer one may input data in the algorithms presented here in order to estimate the Zernike coefficients [23] of the optical aberration contribution of the crystalline lens.

V. REFERENCES

[1] Helmholtz von HH (1909) Handbuch der Physiologishen Optik. In Southall, J.P.C. (Translator), Helmholtz's treatise on physiological optics (pp. 143-172) (1962). New York: Dover. [2] Stone J, The Validity of Some Existing Methods of Measuring Corneal Contour Compared with Suggested New Methods, Brit. J. Physiol. Opt., 1962;19:205-230. [3] Le Grand, Y., El Hage, S. G., Physiological Optics, Springer Series in Optical Sciences, Springer-Verlag, 1980;13. [4] Placido A, Novo Instrumento de Exploração da Cornea, Periodico d’Oftalmológica Practica, Lisboa, 1880;5:27-30. Scheiner C, Sive fundamentum opticum, Innspruk, 1619. [5] Gonzales RC, Woods R. E., Digital Image Processing, Addison-Wesley,1992. [6] Carvalho LAV, Tonissi SA, Romão AC, Santos LE, Yasuoka F, Oliveira AC, Schor P, Chamon W, Castro JC, Desenvolvimento de um Instrumento Computadorizado para Medida do Poder Refrativo da Córnea (Videoceratógrafo), Arq. Bras. Oftal., Dezembro/1998;61(6). [7] Klyce SD, Computer-Assisted Corneal Topography, High Resolution Graphics Presentation and Analyses of Keratoscopy, Invest. Ofthalmol.Vis. Sci.,1984;25:426-435. Liou HL, Brennan NA, Anatomically accurate, finite model eye for optical modeling, J Opt Soc Am A 1997 Aug;14(8):1684-95. [8] Pettit GH, Campin JA, Housand BJ, Liedel KK - Customized corneal ablation: wavefront guided laser vision correction, AUTONOMOUS TECHNOLOGY (ORLANDO, FL) ARVO (The Association for Research in Vision and Ophthalmology), Annual Meeting, Fort Lauderdale, Florida (USA), May 9-14,1999. [9] Thibos LN, The prospects of perfect vision, Journal of Refractive Surgery, Vol. 16, September/October 2000, 540-545. [10] Badouel D, Graphics Gems, volume IV, chapter 7, 1990, 390–393.

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[11] Emsley HH, Visual Optics. Hatton Press Ltd, 5th edition, 1952. [12] J.J Camp, LJ Maguire, e. a., A computer model for the evaluation of the effect of corneal topography on optical performance. American Journal of Ophthalmology, 109(4),379–385, 1990. [13] Kooijman AC, Light distribution on the retina of a wide-angle theoretical eye. J Opt Soc Amer, 73:1544–1550, 1983. [14] Thibos L, Zhang X, Bradley A, (1992). The chromatic eye: a new reduced-eye model of ocular chromatic aberration in humans. Appl Opt, 31(19):3594–3600. [15] Thibos L, Zhang X, Bradley A (1997). Spherical aberration of the reduced schematic eye with elliptical refracting surface. School of Optometry. [16] Nonato LG, Castelo A, Oliveira MCF, A topological approach for handling triangle insertion and removal into two-dimensional unstructured meshes. Cadernos de Computação do ICMCUSP/ São Carlos, 2(3):221–244, 2002. [17] Carvalho LA, Computer Algorithm for Simulation of the Human Optical System and Contribution of the Cornea to the Optical Imperfections of the Eye, Revista de Fisica Aplicada e Instrumentação, vol. 16, no. 1, 2003. [18] Doshi JB, Sarver EJ, Applegate RA, Schematic eye models for simulation of patient visual performance, J Refract Surg 2001 Jul-Aug;17(4):414-9, Erratum in: J Refract Surg 2001 Sep-Oct;17(5):498-9.

[19] Garcia DD, Van de Pol C, Barsky BA, Klein SA, Wavefront Coherence Area for Predicting Visual Acuity of Post-PRK and Post-PARK Refractive Surgery Patients, SPIE conference on Ophthalmic Technologies IX, San Jose, California, January 1999; Proc. SPIE Vol. 3246, p. 290-298, Ophthalmic Technologies VIII, Pascal O. Rol; Karen M. Joos; Fabrice Manns; Eds. ; Publication date: 6/1998.

[20] Brown N (1973) The change in shape and internal form of the lens of the eye on accommodation, Exp. Eye Res., 15, 441-459. [21] Koretz JF, Handleman GH (1986) Modeling age-related accommodative loss on the human eye. Mathematical Modelling, 7, 1003-1014. [22] Koretz JF, and Handelman, G.H. 1988. How the human eye focuses. Scientific American 256 (7): 92-99. [23] Born M, Principles of Optics, Pergamon Press, 1975: 464-466.

VI. AUTHORS

Rodrigo Silva Duran is attending a course of Master Degree in University of São Paulo since 2003, having concluded his graduation in Computer Science in Londrina State University in 2002. His main areas of interest are computer graphics, computer networks and biological models.

Fábio Rodrigues Alves Margarido is student of Computer sciences in the Institute of Mathematical Sciences and Computation of the University of São Paulo, campuses of São Carlos, where it entered 2001. Since August of 2003 project is scholarship holder of scientific initiation for the CNPq developing a computational simulation of the human vision.

Areas of Interest: simulation, computer graphics, computer networks, telecommunications.

Odemir Martinez Bruno acts as professor and researcher in the Institute of Mathematical Sciences and Computation of the University of São Paulo (ICMC-USP) since 2001. It graduated as bachelor in Computer science in 1991. He received the heading from master in physics applied in 1995, working with electronic instrumentation to the Institute of Physics of São Carlos of the University of São

Paulo (IFSC-USP). In 2000 he concluded the PhD in applied physics studying parallelism in natural and artificial vision, in the vision area cybernetics of the IFSC-USP. His main interests in research are: natural and artificial vision, analysis of images, recognition of standards, biotechnology and parallel computation.

Luis Gustavo Nonato received his PhD. degree at Pontifícia Universidade Católica do Rio de Janeiro, Rio de Janeiro, Brazil, in 1998. He has been an Assistant Professor at University of São Paulo, São Carlos, Brazil, since 1999. He currently develops research at the High Performance Computing Laboratory (LCAD) at ICMC - USP. His research

interests involve Computational Geometry, Computer Graphics and Computational Topology. His teaching involves computer graphics, computational mathematics, geometric modelling and computing disciplines.

Luis Alberto Vieira de Carvalho acts as a researcher in (IFSC-USP) and is partner at EYETEC optic equipments LTDA. He received his PhD degree at IF-USP, São Paulo in 2001, working with Applied physics. His main interests areas are: Biomedical engineering, Ophthalmic Instrumentation, Applied Optics, Analytical

Models and of Simulation and Neural networks.