shape modeling and analysis of a human eye based on ... · s. giovanzana et al. shape modeling and...

743

Proceedings of the IMProVe 2011

International conference on Innovative Methods in Product Design

June 15th – 17th, 2011, Venice, Italy

Shape modeling and analysis of a human eye based on individual experimental data

S. Giovanzana (a), G. Savio (a), R. Meneghello (a), G. Concheri (a) (a) DAUR – Laboratory of Design Tools and Methods in Industrial Engineering, University of Padova, Padova 35100, Italy

Article Information

Keywords: Geometric modeling, Ray tracing, Spot of confusion, Virtual eye, Topography.

Corresponding author: Stefano Giovanzana Tel.: +39 049 827 6734 Fax.: +39 049 827 6738 e-mail: [email protected] Address: Via Venezia, 1 35131 Padova - Italy

Abstract

Purpose: The aim of this work is to develop a virtual environment for modeling and analys is an individual virtual eye which is able to integrate the modern imaging techniques as input data.

Method: The virtual environment is developed in a 3D CAD by means of specific plug-ins due to the ability of this software to manage freeform surfaces and to the simplicity in the scripts implementation.

Result: Spot of confusion analysis has been performed on two virtual eyes, with data derived from literature and by topography measurements.

Discussion & Conclusion: Experimental measurements and literature data provide expected results in the spot of confusion analysis.

1 Introduction In the literature there are several eye models based on

population mean values, aimed to quantify the optical, or mechanical properties of the human eye. These models can be classify into two groups: 1) the paraxial schematic eyes are simpler and are only

accurate in the paraxial region; they may be used for simulating basic optical properties, and they are useful for the calculation of entrance and exit pupil [1-3];

2) the finite schematic eyes are often based on paraxial ones, but they are more complex; usually they differ from the others by introducing non-spherical refracting surfaces, gradient index lenses and the components may be aligned arbitrarily. They may be used for simulating more accurately human optics, but different models may be designed for different purpose [4-7].

In order to simplify both the geometric modeling and the analysis of eyes, the models are usually available as 2D profile. Only few authors [8,9] adopt 3D surfaces (e.g. spine) in the shape modeling of eyes, to obtain more accurate models.

On the other hand in the last years many techniques, which are able to survey a 3D topography of corneal and lens surface (e.g. MRI), are developed, but the measurements results are not integrated in the same environment. Consequently no global eye analysis can be performed.

The ability of CADs to represent complex surfaces (by NURBS, spline, mesh, cloud of point etc.) makes these softwares able to support the shape modeling and analysis of the whole eye.

In this work specific plug-ins developed in Rhinoceros 4.0 (McNeel & Associated) are described. These tools allow the shape modeling by NURBS and the analysis by ray tracing, of individual virtual eye, which characteristics are obtained by measurements performed with the recent 3D imaging techniques.

The plug-ins developed are useful tool for the shape modeling of ophthalmic lenses (spectacle, contact and intraocular lenses).

2 Method

2.1 Ray tracing in a 3D environment

Optical analysis of freeform surface is often performed by ray tracing techniques which are able to derive optical properties and spot of confusion [9-11].

The ray tracing method developed for NURBS surfaces is schematically described in Fig. 1 (a): - incident vector i intersect the selected surface - intersection point P is calculated - surface normal at the point P is calculated - generalised Snell’s law is applied - refracted vector i’ is found

This scheme is repeated for every refractive surface encountered along the ray path.

The calculation of the normal for a NURBS surface in the intersection point P follows standard rules of differential geometry [12] and is calculated by the built-in function Rhino.SurfaceNormal.

By the incident vector i, the normal and the refraction indexes n (in the side of the incident ray i) and n’ (in the side of the refracted ray i’), the refracted vector i’ outgoing from the surface can be calculated using the generalised Snell’s law:

744

S. Giovanzana et al. Shape modeling and analysis of a human eye based on individual experimental data

June 15th – 17th, 2011, Venice, Italy Proceedings of the IMProVe 2011

cos'

cos'

'n

ni

n

ni (1)

where angles and are the angles between the incident vector and the normal vector, and between the refracted vector and the normal vector respectively and are calculated by: icos (2)

22

cos1'

1cos co

2

n

n (3)

(a)

(b)

Fig. 1 (a) Flowchart for the calculation of the refracted ray;

(b)results of the implementation of the method.

2.2 Spot of confusion

Fig. 2 (a) shows the flowchart used in the tools for the spot of confusion development.

The rays pattern is formed by 4 rings within a diameter of 8 mm with a distance between rings of 1 mm. The distance between rays within a ring is taken constant and equal to 1 mm.

In Fig. 2 (b) is depicted the path of a complete rays pattern refracted by a single surface; the second surface, which in the eye model simulate the retina, is taken as reference for the image formation and the interception points, located on this surfaces, are defined as spot of confusion P’.

Analysing the spot of confusion P’ barycentre, standard deviation of the points respect to the barycentre and area of the spot are calculated.

2.3 Shape modeling of a virtual eye

The shape of the surfaces of the eye are modelled by NURBS as revolution or free-form surfaces.

2.3.1 Anterior Cornea

The anterior cornea is the dominant refracting surface of the human eye [9]. For the virtual eye, this surface can

either be a theoretical model surface or it can be based on topography measurement.

The most simplest and adopted mathematical function used for modeling the anterior cornea surface is the conicoid. In a 2D plane (e.g. y-z plane) the conicoid curve defined by the equation

22

2

1 yqrr

yz

q1

(4)

in which the radius of curvature r and the aspheric coefficient q. Radius can be experimentally measured by a keratometer. Data from eye models can be also found in the literature and will be used to test the virtual eye.

To obtain the anterior cornea surface, equation 4 is sampled with constant step, interpolated by a curve and rotated around the x-axis (Fig. 3).

(a)

(b)

Fig. 2 (a) Flow chart of the tool developed for the spot of

confusion determination; (b) results representation.

Fig. 3 From left to right: anterior cornea, posterior cornea

and lens model

The majority of eye models found in the literature are rotationally symmetric, but a biconic surface [9] better fit the real shape of a natural eye [13]. In this case or in the case of not regular cornea (which can be measured by topography) the shape can be derived by an alternative method as follow.

Fig. 4 (a) shows the results of a topography measurement (Optikon Keratron), which can be exported in different text files. By the interpretation of this file the elevation map (x-, y-, z-coordinates of points placed along different meridian of the cornea) can be derived and imported in the CAD (Fig. 4 (b)) as a cloud of point.

Ray tracing

Refracted rays, i’

Spot of confusion

Reference surface

Parallel rays, i

Incident vector, i Surface

Intersection point, P(x,y,z)

Surface normal,

Angle, Refracted vector, i’

n n’

Angle , and Snell's law

745



For the ray-tracing procedure is needed to derive a surface (NURBS by Rhinoceros functions or mesh by Delaunay method [14][14,15]) from the cloud of point.

(a)

(b)

Fig. 4 (a) Topographer output data and visualization of

elevation map in dioptres, (b) elevation map in Rhinoceros.

2.3.2 Posterior cornea

Even if the posterior cornea has less effect on the overall refraction of the eye it cannot be neglected. When no data from measurements are available, as in the case of the anterior, the posterior cornea should be represent by a conic which radius can be assumed as 0.81 of the anterior radius [13] (Fig. 3). If the cornea data are imported by a topographer the posterior cornea radius can be taken proportional to the mean anterior radius.

2.3.3 Lens

Chien et al.’s [16] model, which coefficients are derived by imposing geometric constraints [11,17], is adopted geometric modeling of the lens (Fig. 3).

Anterior and posterior radius, aspheric coefficient, diameter, total thickness and ratio between anterior and posterior thickness of the lens are taken as constraints for the model. These coefficients can be derived by lens measurements (e.g. MRI) or by literature data [13,18]. Generally for anterior posterior thickness ratio is taken the value 0.7 [19], or 0.6 [18]. Lens radius, thickness, diameter and position relative to the cornea, change according to age and accommodation [13] and can be considered in the lens modeling stage.

Chien et al.’s model is defined as a 2D curve and consequently the lens surface is obtained by sampling the equation and interpolating the points by a curve and rotating the curve around the x-axis. Refractive index of the lens is taken constant at this stage.

2.3.4 Retina and Sclera

Retina is the imaging surface of the eye, which can be modelled as an aspheric surface [6,18,20] or as an ellipsoid [21]. The geometric model is therefore obtained as in the case of the cornea adopting equation 4.

Sclera is the outer part of the eye and it is not a refractive surface so its shape does not affect the aim of the present study. Sclera is taken as a surface offset out of the retina with a distance from the original surface of 0.5 mm [5].

The complete eye model is represented in Fig. 5.

Fig. 5 Complete virtual eye model

3 Results and discussion Eye surfaces can be derived from literature data that

usually gives the average radius of curvature or derived by measurements (e.g. topographer, scheimpflug lamp, MRI, OWT, OCT [17]).

The modeling and analysis tools have been assessed in two test cases: the first case (C1) performs the Navarro’s data [18]; the second case (C2) assumes the same data substituting the cornea with real Optikon Keratron topographer data (left eye of a man of 42 years old). All the parameters adopted are summarized in Tab. 1.

Adopting this data the ratio between the anterior and the posterior cornea radius of curvature does not respect the value previous cited (0.81 [13]).

In the case C2 the cornea shape (NURBS) is obtained by the patch command of Rhinoceros which need as input data i) sample point spacing, ii) surface u and v spans iii) stiffness; values are detailed in Tab. 2. Adopting these parameters the standard deviation of the clouds of points from the NURBS surface is of 8.089 10-5 mm.

Anterior and posterior thickness of the lens are derived from [18] taking into account the ratio 0.7 found in [19]. Retina radius of curvature is taken equal to 12 mm similar to literature data [21]. Radius and the aspheric coefficient of the retina has not a strong influence in the performed analysis (on-axis analysis). More relevant is the image position for the spot of confusion analysis (e.g. retina) which is fixed as in the Navarro’s model at 24.00398 mm from the cornea apex [18].

Medium / Surface n

(at 589.3 nm) r [mm] q [mm]

t [mm]

Air 1 /

Cornea Anterior C1

1.376 7.72 -0.26

0.55 Anterior C2 Topographer data Posterior 6.5 0

Aqueous 1.3374 / 3.05

LensAnterior

1.42 10.2 -3.1316 1.647

Posterior -6.0 -1 2.353 Vitreous 1.336 / 16.4 Retina / -12 0 /

Tab. 1 Parameters used for shape modeling of virtual eye.

Sample point spacing [mm]

u spans

v spans

Stiffness

NURBS surface 0.31 50 50 1

Tab. 2 Data used for the NURBS creation

746



Using the method developed it is immediate to evaluate the intersections between the refracted rays and the retina surface (i.e. spot of confusion). Fig. 6 represents the path of the pattern of rays in the eye as seen in Rhinoceros.

Fig. 6 Parallel rays refracted into the virtual eye model

Fig. 7 (a) presents the spot of confusion of the case C1. As it can be seen the spot is regular and the rays remains in a circumference which radius is related to the spherical aberration.

(a)

(b)

Fig. 7 (a) Spot of confusion for the case C1 and

(b) for the case c2.

Once the anterior cornea surface was substituted with the real cornea data (case C2) the interceptions between the refracted rays and the retina surface are calculated and represent in Fig. 7 (b). It can be seen that the 4 rings does not preserve their shape with more distortion when the ring increase its radius. More in detail the shape of the rings appears similar to an ellipse which means that the cornea is astigmatic; this result is due to a physiological condition: radius of curvature of the anterior surface is usually greater in the horizontal meridian than that in the vertical meridian [13] (“with the role” astigmatism), so that the horizontal power of the cornea is 43.73 D and the vertical power of the cornea is 44.95 D. In Fig. 7 (b) it can

be seen then that the points in the y-direction are much spread then the x-direction due to the fact that they are focalized before into the eye.

Tab.3 collects centre, standard deviation and area of both the spots of confusion. Since the virtual eye model is created as rotation symmetry along the z-axis the centre is exactly in the origin of the xy-plane. For the virtual eye of the case C2, an imbalance in the coordinates of the centre of the spot appears which can be related to the to the real visual axis of the eye and to the real location of the fovea [13].

Area and standard deviation of the spot of confusion take a double or more value in the case C2.

As stated in [22], the introduction of an aspheric coefficient or of an ellipsoidal shape [21] for the retina may improve the performance of the virtual eye in the out-axis analysis, obtaining a more realistic model.

Case Centre [mm]

Standard deviation [mm]

Area [mm2]

C1 (0,0) 0.0702 0.0934 C2 (-0.0046,-0.0014) 0.1782 0.1978

Tab. 3 Centre, standard deviation and area of the spot of confusion for the test cases.

In this work the pupil diameter was taken equal to 8 mm according to other works [1-9,18-22]. Reducing the pupil diameter the ray pattern dimension can be reduced; consequently the spot of confusion dimension is reduced. For example adopting a 6 mm diameter pupil a spot of confusion with standard deviation equal to 1.435 mm and area equal to 0.1159 mm2 have been found in the case C2.

4 Conclusion In this work several methods for shape modeling and

analysis of a virtual eye, with real characteristics derived by measurements, has been proposed.

Rhinoceros has been adopted as development environment for specific plug-ins implementation due to the ability of the CAD to manage NURBS surfaces and to the simplicity in the scripts implementation.

Model derived from different works and measurements detected by topography are together develop and analyzed.

The plug-ins is suitable in the ophthalmic lens design and analysis as in the real eye refractive diseases evaluation.

Acknowledgement The authors wish to thank G. Guerra for the topography

data provided.

References [1] A. Gullstrand. Helmholz’s handbuch der

physiologichen optic. Southall edition. 1909. [2] H.H. Emsley. Visual optics. Butterworths edition. 1952. [3] R.B. Rabbetts. Bennett and Rabbetts’ clinical visual optics. Butterworth-Heinemann 3rd edition. 1998. [4] H.L. Liou and N.A. Brennan. Anatomically Accurate, Finite Model Eye for Optical Modeling. Journal of the Optical Society of America A 14-8 (1997) pp. 1684-1695. [5] S. Norrby. The Dubbelman eye model analysed by ray tracing through aspheric surfaces. Ophthalmic and Physiological Optics 25 (2005) pp. 153-161.

-0.2

-0.1

0

0.1

0.2

-0.2 -0.1 0 0.1 0.2

x [mm]

y [m

m]

Virtual eye

-0.4

-0.2

0

0.2

0.4

-0.4 -0.2 0 0.2 0.4

x [mm]

y [m

m]

747



[6] D.A. Atchison. Optical Models for Human Myopic Eyes. Vision Research 46 (2006) pp. 2236-2250. [7] A.V. Goncharov and C. Dainty. Wide-field schematic eye models with gradient-index lens. Journal of Optical Society of America A 24 (2007) pp. 2157-2174. [8] B. Tan. Optical Modeling of Schematic Eyes and Ophthalmic Applications. University of Tennessee, Noxville. 2008. [9] J. Einighammer. The Individual Virtual Eye. University of Tubingen. 2009. [10] G. Savio, G. Concheri, R. Meneghello. Simulazione delle proprietà ottiche e progettazione di lenti mediante tecniche di ray tracing. XVI ADM - XIX INGEGRAF "Dall'idea al prodotto: la rappresentazione come base per lo sviluppo e l'innovazione", Firenze 6-8 Giugno (2007) pp 259-266. [11] S. Giovanzana. A virtual environment for modeling and analysis of human eye. Università di Padova. 2011. [12] M.E. Mortenson. Geometric Modeling. Industrial Press, Inc. New York 3rd edition. 2006. [13] D.A. Atchison. Optics of the human eye. Butterworth-Heinemann 2003. [14] M. De Berg, M. Van Kreveld, M. Overmars, O. Schwarzkopf. Computational Geometry Algorithms and Applications. Springer-Vaelag, 1997. [15] Pointset Reconstruction, from Rhino 4.0 Labs Tools: http://wiki.mcneel.com/labs/pointsetreconstruction. Accessed 04 Feb 2011. [16] M.C. Chien, H. Tseng, R.A. Schachar. A mathematical expression for the human crystalline lens. Comprehensive Therapy, 29-4 (2003) pp245–258. [17] S. Giovanzana, G. Savio, R. Meneghello, G. Concheri. Shape analysis of a parametric human lens model based on geometrical constraints. Journal of Modern Optics, in press. [18] R. Navarro, J. Santamaría, J. Bescós. Accommodation-dependent model of the human eye with aspherics. Journal of the Optical Society of America A 2, 8 (1985) pp. 1273-1281. [19] M. Dubbelman, G.L., Van der Heijde, H.A., Weeber. The thickness of the aging human lens obtained from corrected Scheimpflug images. Optometry and Vision Science 78 (2001) pp. 411-416. [20] A.C. Kooijman. Light distribution on the retina of a wide-angle theoretical eye. Journal of the Optical Society of America 73 (1983) pp. 1544-1550. [21] D.A. Atchison, N. Pritchard, K.L. Schmid, D.H. Scott, C.E. Jones, and J.M. Pope. Shape of the Retinal Surface in Emmetropia and Myopia. Investigative ophthalmology and visual science 46-8 (2005) pp. 2698-2707. [22] R.C. Bakaraju, K. Ehrmann, E. Papas, A. Ho. Finite schematic eye models and their accuracy to in-vivo data. Vision Research 48 (2008) pp. 1681-1694.

shape modeling and analysis of a human eye based on ... · s. giovanzana et al. shape modeling and...

Documents