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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 25 — Dec. 8, 2008
  • pp: 20490–20502
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Modeling the eye’s optical system by ocular wavefront tomography

Xin Wei and Larry Thibos  »View Author Affiliations


Optics Express, Vol. 16, Issue 25, pp. 20490-20502 (2008)
http://dx.doi.org/10.1364/OE.16.020490


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Abstract

Purpose: Ocular wavefront tomography (OWT) is the process of using wavefront aberration maps obtained along multiple lines-of-sight (LoS) to determine the shape and position of the major refracting elements of an eye. One goal of OWT is to create a customized schematic model eye that is anatomically similar and functionally equivalent to the individual eye over a large field of view. Methods: Wavefront aberration maps along multiple LoS were used as design goals for configuring a generic, multisurface model eye with aberrations that match the measurements. The model was constrained by gross anatomical dimensions and optimized to mimic the measured eye. The method was evaluated with two test cases: (1) a physical model eye with a doublet lens measured with a clinical wavefront aberrometer along six LoS between -31 deg and +29 deg eccentricities, and (2) a mathematical model of the myopic eye for which wavefront aberrations were computed by ray tracing. Results: In case 1, the OWT algorithm successfully predicted the structure of the doublet model eye from the experimental on- and off-axis aberration measurements. In case 2, the algorithm started with a symmetric five surface model eye and optimized it to generate the on- and off-axis aberrations of a GRIN myopia model eye. The adjusted model closely mimicked the physical parameters and optical behavior of the expected myopia model eye over a large field of view. The maximum discrepancy between aberrations of the OWT optimized model and measurements was 0.05 microns RMS for test case 1 and 0.2 microns RMS for test case 2. Conclusion: Our implementation of OWT is a valid, feasible, and robust method for constructing an optical model that is anatomically and functionally similar to the eye over a wide field of view.

© 2008 Optical Society of America

1. Introduction

A conceptually different approach to ocular tomography treats the problem as a case of customized modeling that can be approached using tools developed in the field of optical design. The idea is to construct a model eye incorporating pre-measured biometric data (e.g. surface profiles from cornea topography) so that the wavefront aberrations of a customized eye model are similar to measured wavefront aberrations. This customization process was used initially to develop an axial eye model from wavefront aberrations [6

6. R. Navarro, L. Gonzalez, and J. L. Hernandez-Matamoros, “On the prediction of optical aberrations by personalized eye models,” Optom Vis Sci 83, 371–381 (2006). [CrossRef] [PubMed]

, 7

7. A. V. Goncharov and C. Dainty, “Wide-field schematic eye models with gradient-index lens,” J. Opt. Soc. Am. 24, 2157–2174 (2007). [CrossRef]

] but more recently this approach has been extended to model off-axis aberrations of the near peripheral visual field [8

8. A. V. Goncharov, M. Nowakowski, M. T. Sheehan, and C. Dainty, “Reconstruction of the optical system of the human eye with reverse ray-tracing,” Opt. Express 16, 1692–1703 (2008). [CrossRef] [PubMed]

]. The result is an optical model that mimics the measured wavefront aberrations along multiple line-of-sight (LoS) over a moderate field of view for which the eye’s entrance pupil is approximately circular. A major goal of the present study was to extend this approach into the wide field domain where the entrance pupil is elliptical with an aspect ratio that varies with field angle.

The present paper develops a conceptual framework called ocular wavefront tomography (OWT) for revealing the eye’s internal optical structure. Our approach converts the optical tomography problem into a well-formulated optimization problem for which the dual concepts of functional equivalence and anatomical similarity become objective standards for judging results. Within this framework, the researcher plays the role of an optical designer whose goal is to determine that configuration of a template optical system which best satisfies optical performance specifications, subject to physical constraints imposed by the template’s optical components. In OWT, optical performance is specified by a series of wavefront aberration maps that describe the optical aberrations of the eye measured for a series of field angles (i.e. along peripheral LoS) covering as much of the eye’s field of view as required for an envisioned application. The physical constraints are the gross anatomical dimensions common to all eyes or, if available, the specific dimensions measured for an individual eye. Depending on the intended application, the template model might contain a single refracting surface, or four refracting surfaces (cornea + lens), or gradient index (GRIN) descriptions of the eye’s dioptric components. Taken together, the measured wavefront aberrations and the anatomical constraints are used as merit functions that define an optimization goal in the optical design process. A successful outcome produces the optimum configuration of the chosen template optical system for mimicking the eye’s behavior over the full visual field. To the extent that the template model incorporates the essential features of the human ocular system required for the intended application, the resulting eye model may be interpreted as an anatomically accurate schematic representation of the individual eye’s optical system.

In this paper, we evaluate the feasibility and validity of our proposed OWT framework using two test cases. In test-case #1, the off-axis wavefront aberrations of a commercial doublet lens with known parameters was characterized by numerical ray tracing, and also by experimental measurements with a Hartmann-Shack wavefront sensor, over a large field-of view (from -31 deg to 29 deg) taking into account the elliptical distortion of the pupil. We learned that when only a few parameters of the lens were free variables for optimization, OWT correctly determined their true values uniquely as the only values that could account for the full set of wavefront aberration measurements. However, for a larger number of unknown free-parameters, incorporating the physical constraints into the optimization goal was necessary for the OWT method to determine the true values. In test case #2, we generated wavefront aberration functions along multiple LoS for an optical model of the eye that employs GRIN optics [9

9. D. A. Atchison, “Optical models for human myopic eyes,” Vision Res. 46, 2236–2250 (2006). [CrossRef] [PubMed]

]. These data represent the aberration structure of an eye that we wish to model using a totally different template schematic eye. The selected template was an anatomically accurate, wide-angle model of the eye with homogeneous-index optics developed by Escudero-Sanz et al [10

10. I. Escudero-Sanz and R. Navarro, “Off-axis aberrations of a wide-angle schematic eye model,” J. Opt. Soc. Am. 16, 1881–1891 (1999). [CrossRef]

]. We found that OWT successfully re-configured the Escudero-Sanz model so that its optical properties closely matched the Atchison model, even without the benefit of GRIN optics. This result demonstrates that OWT can produce a functional model that is anatomically realistic and that reproduces specified optical behavior over a wide field of view. However, the solution was not unique because of the large parameter space available for the optimization. To achieve a unique solution requires (1) more off-axis wavefront measurements of the tested eye and (2) tighter constraints on free parameters that might be provided by additional data on the shape and location of the individual eye’s major refractive components.

2. Methods

2.1 Elliptical entrance pupil

Ocular aberrations are specified with respect to the eye’s entrance pupil, which is the image cast by the cornea of the circular aperture in the iris. When viewed from the perspective of wavefront sensors along a secondary LoS (Fig. 1(a)), the entrance pupil becomes elliptical. For example, when we used a Hartmann Shack wavefront sensor to measure wavefront slopes off-axis for a physical model eye, the grid of sample points was confined to the interior of the elliptical entrance pupil. Several methods are available for retrieving wavefront phase from gradient measurements over an elliptical pupil [11

11. H. Sumita, “Orthonormal expansion of the aberration difference function and its application to image evaluation,” Jpn. J. Appl. Phys. 8, 1027–1036 (1969). [CrossRef]

16

16. D. A. Atchison, D. H. Scott, and W. N. Charman, “Measuring ocular aberrations in the peripheral visual field using Hartmann-Shack aberrometry,” J. Opt. Soc. Am. 24, 2963–2973 (2007). [CrossRef]

]. We used the scaling method described by Sumita [11

11. H. Sumita, “Orthonormal expansion of the aberration difference function and its application to image evaluation,” Jpn. J. Appl. Phys. 8, 1027–1036 (1969). [CrossRef]

] and Atchison [13

13. D. A. Atchison and D. H. Scott, “Monochromatic aberrations of human eyes in the horizontal visual field,” J. Opt. Soc. Am. 19, 2180–2184 (2002). [CrossRef]

, 16

16. D. A. Atchison, D. H. Scott, and W. N. Charman, “Measuring ocular aberrations in the peripheral visual field using Hartmann-Shack aberrometry,” J. Opt. Soc. Am. 24, 2963–2973 (2007). [CrossRef]

] that we implemented in Matlab.

Fig. 1. Geometry of the eye’s entrance pupil for off-axis viewing along a secondary line-ofsight. (a) A cross-section of the cornea and pupil in the plane formed by the primary and secondary lines-of-sight. Wavefront aberrations measured with respect to the x’,y’ plane of the secondary entrance pupil are transferred mathematically to the x,y plane of the primary entrance pupil. (b) Measurement axis of the aberrometer is aligned with a secondary line-ofsight (bold arrow) by rotating the instrument by the angle ε in a plane inclined by angle θ with the horizontal. (c) When projected into the x,y plane of the primary entrance pupil, the x’,y’ plane of the secondary entrance pupil is tilted by angle θ. The secondary entrance pupil is elliptical with minor and major axes aligned with the x’ and y’ axes, respectively. The aspect ratio of the elliptical pupil (i.e. minor axis/major axis) equals cos ε.

We also generated test data using numerical ray tracing of eye models. Most ray tracing algorithms (e.g. Zemax ray tracing engine) calculate optical path difference (OPD) of rays defined over an isotropic grid of (x,y) sample points in the plane of the primary (i.e. on-axis) entrance pupil. Therefore, to compare the measured wavefront along the secondary LoS with the ray tracing prediction, the measured wavefront over the secondary pupil plane (x’,y’) had to be remapped to the iris aperture coordinates. The geometry of this mapping of entrance pupil coordinates to iris aperture coordinates is shown in Fig. 1, where every point (x,y) in the circular domain of the iris aperture maps uniquely to a point (x’,y’) in the entrance pupil. We assume the primary LoS is perpendicular to the iris plane and that the secondary entrance pupil is perpendicular to the secondary LoS (Fig. 1). Since an entrance pupil is the image of the physical pupil cast by the cornea, the centers of primary and secondary entrance pupils both map to the center of the physical pupil. Consequently, the mathematical transformation of coordinates between the secondary entrance pupil and primary entrance pupil is described by a rotation and scaling matrix in Eq. (1) At large eccentricities Eq. (1) becomes inaccurate because the secondary entrance pupil will deviate from an elliptical shape due to aberrations of cornea refraction.

(x'y')=(cosθsinθsinθcosθ)(cosε001)(cosθsinθsinθcosθ)(xy)
(1)

A secondary LoS is established for eyes by rotating the aberrometer by an amount ε (eccentricity) about an axis in the iris plane that is inclined at the angle θ (meridian) to the horizontal, as illustrated in Fig. 1(b). The relationship between the rotation angles (ε,θ) of the instrument and the direction cosines (l,m,n) of the secondary LoS is:

tanη=ml,tanε=m2+l2n
(2)

2.2 Merit function for optimization

Once the off-axis wavefront aberrations are measured and remapped to the iris aperture plane, a merit function that numerically represents the optimization goal must be defined. In our case, the goal of the OWT algorithm is to reconstruct a customized wide angle schematic eye that is anatomically similar and functionally equivalent to the tested eye. A successful merit function will be a good measure of these two objectives. The first part of the merit function represents the anatomical similarity between a template model eye based on biometric data from a population of eyes and the customized model produced by the optimization engine. For example, by representing the structural parameters of the template model and customized model as two vectors, a distance measure (we used Euclidean distance) between these two vectors is a measure of anatomical similarity. The second part of the merit function measures the difference between the remapped off-axis wavefront measurements and the ray tracing prediction of the customized model. For example, the average RMS of wavefront differences along the multiple LoS is a convenient measure that we adopted. Since the off-axis wavefront spanning the elliptical entrance pupil is remapped to the circular iris aperture, the remapped wavefront measurements and ray tracing prediction are both represented by a vector of coefficients for Zernike circle polynomials. The Euclidean distance between these two Zernike vectors was taken as a measure of the functional equivalence of the model and the test eye that produced the off-axis aberration data. In practice, the formulation of the merit function can be very flexible. The off-axis wavefront can be formulated in many different formats [11

11. H. Sumita, “Orthonormal expansion of the aberration difference function and its application to image evaluation,” Jpn. J. Appl. Phys. 8, 1027–1036 (1969). [CrossRef]

, 12

12. F. Roddier and C. Roddier, “Wavefront reconstruction using iterative Fourier transforms,” Appl. Opt. 30, 1325–1327 (1991). [CrossRef] [PubMed]

, 14

14. L. Lundstrom, P. Unsbo, and J. Gustafsson, “Off-axis wave front measurements for optical correction in eccentric viewing,” J. Biomed. Opt. 10, 034002 (2005). [CrossRef] [PubMed]

16

16. D. A. Atchison, D. H. Scott, and W. N. Charman, “Measuring ocular aberrations in the peripheral visual field using Hartmann-Shack aberrometry,” J. Opt. Soc. Am. 24, 2963–2973 (2007). [CrossRef]

]; different components can be weighted differently as required. For example, if the accuracy of modeling the on-axis spherical aberration of a tested eye is the highest priority during the customized modeling, more weight can be assigned to the on-axis spherical aberration during the formulation of the merit function. The only essential requirement of the merit function is that it should embody the dual concepts of anatomical similarity and functional equivalence.

Having formulating the merit function in the foregoing manner, the tomography problem of adjusting a template model to become anatomically similar and functionally equivalent to the tested individual eye is converted into an optimization problem of finding a customized model eye that achieves a global minimum of the merit function. Many practical approaches in the field of artificial intelligence such as damped-least squares, simulated annealing, neural networks, case-based reasoning, and expert systems are available to solve this optimization problem.

To compare wavefront measurements over the elliptical pupil with Zemax ray tracing predictions, we used “Zemax extensions” [17

17. Zemax User Guide (Zemax Development Corporation, 2006).

] to customize a merit function based on the mapping geometry described in section 2.1. After defining a functional equivalence measure, we set a series of boundary operands [17] to constrain the optimized model eye within the normal range of general ocular shape. These boundary constraints are regarded as the anatomical similarity measure in the merit function. After formulating the OWT framework in Zemax, the optimization engine in Zemax is used to iteratively adjust the customized model to meet the dual concepts of anatomical similarity and functional equivalence. Although we implement this OWT framework in Zemax, the OWT framework is independent of the choice of optical design software. To implement the OWT framework requires (1) specifying off-axis wavefront aberrations over multiple LoS, (2) comparing those aberrations with ray-tracing predictions, and (3) formulating the optical tomography goals as optimization problem. These features can be implemented with various optical design programs or customized optimization engines.

2.3. Test cases

To verify the validity, feasibility, and robustness of our implementation of OWT, the following two test cases were used in numerical simulations and experiments.

Test case #1: A physical optical system consisting of an aperture and a doublet lens (LAO 134 doublet from Melles Griot Inc.) was measured with a commercial Hartmann-Shack aberrometer (COAS, Wavefront Sciences, Inc.) aligned initially with its measurement axis coincident with the optical axis of the lens. The aberrometer was then rotated by an angle ε (eccentricity) about a vertical axis in the plane of the aperture and passing through the center of the aperture to obtain measurements along a secondary LoS lying in the horizontal meridian (θ=0 degree). Wavefront measurements were obtained along multiple LoS (-31, −21, -11, 9, 19, 29 degrees) for a 5 mm on-axis entrance pupil. We also computed the expected wavefronts and vector of Zernike coefficients for these LoS by using the ray tracing engine in Zemax. The template model for the Melles Griot lens was the 7-parameter doublet shown in Fig. 2, but with multiple manufacturer’s parameters altered significantly (e.g. a factor 2 change in radius of curvature or thickness). The OWT implementation was then used to retrieve those altered parameters.

Fig. 2. Schematic diagram of the tested doublet. The goal of ocular wavefront tomography is to recover the 7 parameters that characterize the lens (3 radii of curvature, 2 thicknesses, 2 indices of refraction) based on wavefront maps measured along 6 lines of sight.

Test case #2: We simulated experimental data by computing wavefront aberration maps for a computer implementation of the Atchison GRIN myopia model eye [9

9. D. A. Atchison, “Optical models for human myopic eyes,” Vision Res. 46, 2236–2250 (2006). [CrossRef] [PubMed]

]. Ray tracing was performed by Zemax for a circular aperture of the iris using a fixed grid of sample points that was the same for all secondary LoS. Thus a given ray originating from a fixed normalized field point, and passing through point (x,y) in the iris plane, also passed through point (x’,y’) in the entrance pupil and was associated with an OPD error that depended on the direction cosines (l,m,n) of the line-of-sight. For each aberration map associated with a given LoS, Zernike aberration coefficients were computed to the 5th order over the circular domain of the iris aperture and the zero-order coefficient (piston) and first-order coefficients (tip and tilt) were discarded before setting up a merit function in Zemax. To perform the optimization, parallel rays from an infinitely distant object point were traced through the model eye to its exit pupil. Wavefront error was computed as the optical path difference between the exiting wavefront and a reference sphere centered on the intersection point of the chief ray with the retinal surface.

3. Results

3.1. Test case #1: doublet model eye

Our first test of OWT was to determine if the method could discover the manufacturer’s specified value of a single lens parameter when the merit function was based solely on theoretical values of Zernike aberration coefficients Z22 (astigmatism) and Z31 (coma) obtained by numerical ray tracing of the lens. Each of the 7 lens parameters was tested in turn and the results (cases 1–7 of Table 1) indicated nearly perfect recovery of the manufacturer’s specification. Next, we tested groups of parameters (radii, thickness, or index of refraction) and again found nearly perfect recovery of the specified values (cases 8–10). However, large errors were made when 5 of the 7 lens parameters in the nominal starting model needed significant adjustment (case 11). This failure was traced to a local-minimum condition that prevented the discovery of a global minimum because of the increasing complexity of the optimization space. The optimization space was reduced by adding additional anatomical constraints, which resulted in a successful recovery of the structural parameters of the lens with less than 1% error (test case 12). These additional constraints may be interpreted as a method for enforcing anatomical similarity by not allowing parameters to become unreasonable, or non-physiological.

Table 1. Results of using OWT to retrieve the structural parameters of a doublet lens from simulated measurements of wavefront aberrations calculated by ray tracing along different LoS.

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The second test of the OWT method used a merit function based on measured wavefront aberrations obtained by a clinical Hartmann-Shack Wavefront aberrometer (COAS) on a physical lens. Anatomical similarity was enforced, and the optimization space was reduced, by the additional constraint that parameter values could not differ by more than twofold from the manufacturer’s specification. The initial parameter values of 5 lens parameters specifying radii and separation of all 3 surfaces were adjusted significantly (Table 2) from the manufacturer’s specification. This procedure was sufficient to produce large errors in the aberration maps of the starting model compared to measured wavefront aberrations along the multiple LoS (upper row of aberration maps in Fig. 3(b)). These errors are caused by astigmatism as well as coma aberrations (Fig. 3(a)).

Table 2. Comparison table of physical parameters

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The outcome of OWT is shown in the bottom row of aberration maps in Fig. 3(b). Each of these maps is the difference between the aberration of the OWT model and the theoretical expectations. With one exception (-21 deg, where we suspect a possible measurement error) each of these maps is very flat, indicating the model determined by OWT is functionally equivalent to the theoretical model based on manufacturer’s specifications. Figure 3(a) summarize the RMS change between retrieved Z22 and Z31 Zernike coefficients over the circular pupil before and after applying OWT relative to the COAS measurements. Corresponding wavefront difference maps over the actual elliptical entrance pupils along the multiple LOS are summarized in Fig. 3(b). The RMS differences reduced to within the level of 0.01 microns for all LoS. Table 2 shows the structural similarity of the optimized model after OWT from the initial template relative to the vendor’s specification. Other than the radius of the last surface (r3), all other restored parameters differed from manufacturer’s specification by less than 1% (6% for r3).

Fig. 3. Outcome of OWT for the doublet test case. (a) Error in Zernike coefficients for the starting model (open symbols) and the final model produced by OWT (filled symbols). (b) Wavefront difference between starting model and theoretical model (upper row) and difference maps between the final model and the theoretical model (bottom row).

3.2 Test case #2: schematic eyes

The doublet test demonstrated that OWT is able to reconfigure a generic model of a doublet lens with starting parameters that are reasonable, but significantly incorrect, into a model that is nearly identical to the manufacturer’s specification. Our second test case was designed to test the validity of OWT for human eyes, which have an optical structure that has been modeled by two well-known schematic eyes: Escudero-Sanz wide angle schematic (WAS) eye [10

10. I. Escudero-Sanz and R. Navarro, “Off-axis aberrations of a wide-angle schematic eye model,” J. Opt. Soc. Am. 16, 1881–1891 (1999). [CrossRef]

] and Atchison’s myopia model (MYO) eye [9

9. D. A. Atchison, “Optical models for human myopic eyes,” Vision Res. 46, 2236–2250 (2006). [CrossRef] [PubMed]

]. Both models purport to represent optical behavior over a wide field of view, but have very different peripheral aberrations. One major structural difference between the two models is that WAS assumes the refractive index of the crystalline lens is homogeneous, whereas the MYO model includes a gradient index (GRIN) profile. Another major difference is that WAS is a symmetric model that assumes the primary LoS coincides with the model’s optical axis, whereas MYO is an asymmetric model. Our strategy in testing the OWT method was to use WAS as an anatomically reasonable template model. The structural parameters of WAS were then adjusted in an attempt to make it behave the same as MYO.

Fig. 4. Structural changes in the WAS model to meet MYO model’s optical performance from - 30 degree to +30 degree horizontally.

Test data for the MYO eye were simulated by computing wavefront aberrations (6mm entrance pupil, 555nm light) along 7 different LoS (-30, -20, -10, 0, 10, 20, 30 deg). Zernike representations of these 7 aberration maps served as the optical performance goal in the merit function. Anatomical features of the MYO eye were used to formulate additional constraints in the merit function to ensure anatomical similarity. The results, shown in Fig. 4 and Table 3 showed a reasonable agreement of anatomical similarity of the optimized model (OWT) to the myopia eye model (MYO). Furthermore the equivalent power of the optimized OWT model eye was 62.00 D, which was similar to MYO model. On-axis spherical aberration for 6mm diameter pupil was achieved to be exactly the same as MYO model (0.09 microns).

Table 3. Comparison table of physical parameters among WAS model, OWT model, and MYO model.

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[a] The MYO model was listed incompletely (e.g. GRIN lens, biconic retina) for comparison purpose.

[b] The WAS model is a symmetric model. The retina in OWT model was laterally shifted and tilted to compensate the asymmetric feature of MYO model.

Besides achieving experimental on-axis spherical aberration and anatomical similarity, the off-axis optical quality in object space agreed with MYO model as in Fig. 5. The RMS of wavefront error difference between WAS model and MYO model was reduced within a level of 0.2 microns after applying the OWT algorithm.

Fig. 5. Outcome of OWT for the schematic eyes test case. (a) RMS difference between the original WAS model and MYO model (open symbols) is compared with RMS difference between the modified WAS model and MYO model (filled symbols). (b) Wavefront difference between the original WAS model and MYO model (upper row) and difference maps between the modified WAS model and MYO model (bottom row).

The fact that a 5-surface model with homogeneous optics can be found that mimics reasonably well the behavior of a GRIN model over a wide field does not imply the OWT method for finding the model lacks sensitivity. It simply means that including the complexity of GRIN optics in the model is not necessary for reproducing adequately the effect of GRIN optics on wavefront aberrations along the seven specified LoS. Presumably an even better description would have been obtained had we used a GRIN template, but clearly there is not a lot of room for improvement in this particular example.

4. Discussion

Successful implementation of OWT using the optical design program Zemax depends on two primary factors: the initial conditions of the template model to be optimized and the design of the merit function. The merit function has two parts that correspond to anatomical similarity and functional equivalence. Anatomical similarity is established by the form of the template model (e.g. a 3 refracting surface model is used to describe a doublet lens), the initial values assigned to parameters of the template model (e.g. surface curvatures, spacing, and refractive indices), and out-of-range limitations for the parameters. We assert at the outset that if the optimization engine is able to return a solution within the bounds established by these constraints then, by agreement, the optimized model is anatomically similar to the initial model. The degree of anatomical similarity required of the model will be determined primarily by its intended application. If the basic structure of the optical model and the permitted range of parameter values are in agreement with our knowledge of the human eye, then we are justified in concluding that optimized model is anatomically accurate.

By envisioning OWT as an optimization problem, many optimization tools in the field of optical design can be used to find a solution. Optical designers create lenses to meet specific optical performance requirements for specific applications, subject to manufacturability constraints. Similarly, the vision scientist creates eye models that mimic the measured optical performance of real eyes. This is done for a specific purpose, or application, and is subject to anatomical constraints. Since every eye is a unique, complex optical system, converting this OWT framework into a well-formulated optimization problem is not trivial. Designing an optical model to behave like a human eye is not only a science but also an art that benefits from experience. In practice, successful optical designs result from the skill and experience of the optical designer and are not necessarily unique. The same holds for optimizing the template model in OWT. Similarly, the basic principles of optical design can be adapted for OWT as follows: (1) set a clear design objective, (2) balance anatomical correctness against functional equivalence, (3) choose a template that matches the application. Each of these principles is elaborated below.

Firstly, a clear designing objective is necessary. In our 2nd test case, for example, our goal was to modify the Escudero-Sanz WAS model eye (a symmetric five surface model) to become functionally equivalent to Atchison’s MYO model (an asymmetric model with GRIN optics). To stress this functional equivalence over a large filed of view, we evaluated wavefront aberrations along 7 different LoS within +/-30 deg. To stress anatomical similarity, we incorporated a series of constraints (boundary operands in Zemax) based on anatomical data from the literature [9

9. D. A. Atchison, “Optical models for human myopic eyes,” Vision Res. 46, 2236–2250 (2006). [CrossRef] [PubMed]

]. These anatomical constraints included the apical curvature and conic constant of the major refracting surfaces, thicknesses of the anterior chamber, lens and posterior chamber, decentration of the fovea, tilt and decentration of the retina, and the effective focal length and axial length of the eye. Formulation of the merit function with these design objectives was essential for success (the model defined in Table 3). In future applications, we will use this method to customize a wide-angle model eye that functionally and anatomically mimics the individual human eye. Depending on the desired complexity of the model (e.g. GRIN vs. homogeneous crystalline lens), we will need more or less information about the tested eye. To achieve functional equivalence, it might be necessary to measure a large series of wavefront aberrations over an extensive, two dimensional visual field. To achieve anatomical similarity, it may become necessary to utilize biometric data (e.g. cornea shape) obtained either from statistical population data or from auxiliary measurements (e.g. cornea topography) to limit or simplify the optimization domain.

Secondly, good optical design achieves a balance between the required optical performance, tolerance, and optomechanical constraints [18

18. R. Shannon, The Art and Science of Optical Design (Cambridge Univ Press, 1997).

, 19

19. M. Kidger, Intermediate Optical Design (SPIE, 2004). [CrossRef]

]. Similarly for model eye customization, it is necessary to balance anatomical correctness against functional equivalence in a way that is appropriate for a specified application. For example, in the presence of typical measurement noise (e.g. an outlier along LoS at -21 degree shown in Fig. 2(a)), we tightened the anatomical similarity constraints (increasing the boundary operands weights) until off-axis wavefront aberrations of the optimized model along most LoS matched the wavefront aberration measurements. Balancing competing anatomical constraints is also important. Different ocular parameters have different sensitivities to the wavefront measurements over a large field of view. Parameters with relatively high sensitivity are more easily determined by the wavefront aberration measurements. Therefore, we assign a relatively low weight to those sensitive parameters. The weights of the anatomical constraints are iteratively adjusted in this way until the customized model eye becomes a balanced design to meet the dual design objectives of functional equivalence and anatomical similarity.

Lastly although the OWT framework is able to convert the optical tomography into an optimization problem, the method itself is independent of the choice of initial template model. In the field of optical design, simplicity is often the highest priority. Likewise, a reduced-eye model with a single refracting surface may be very attractive, for example, when designing clinical instrumentation that will be optically coupled to the eye (e.g. a fundus camera, or wide-angle spectacle lenses [20

20. G. Smith, D. A. Atchison, C. Avudainayagam, and K. Avudainayagam, “Designing lenses to correct peripheral refractive errors of the eye,” J. Opt. Soc. Am. 19, 10–18 (2002). [CrossRef]

]). A scientific study of accommodation, on the other hand, would require a template model with additional refracting surfaces, or possibly a GRIN element, representing the crystalline lens [6

6. R. Navarro, L. Gonzalez, and J. L. Hernandez-Matamoros, “On the prediction of optical aberrations by personalized eye models,” Optom Vis Sci 83, 371–381 (2006). [CrossRef] [PubMed]

]. Thus, the intended application of the model should guide the selection of the template model to be optimized within the OWT framework.

This flexibility in selecting the template model makes OWT a powerful tool for simplifying a model eye’s anatomical structure while preserving its functional behavior. For example, we used OWT to configure the Indiana-eye template [21

21. L. N. Thibos, M. Ye, X. Zhang, and A. Bradley, “Spherical aberration of the reduced schematic eye with elliptical refracting surface,” Optom Vision Sci. 74, 548–556 (1997). [CrossRef]

, 22

22. Y. Z. Wang and L. N. Thibos, “Oblique (off-axis) astigmatism of the reduced schematic eye with elliptical refracting surface,” Optom Vis Sci 74, 557–562 (1997). [CrossRef] [PubMed]

] (consisting of a single refracting surface, a pupil, and curved retinal surface) to mimic the behavior of the Escudero- Sanz wide angle schematic (WAS) eye with four refractive surfaces. The method returned a configuration of the Indiana eye (Fig. 6) with off-axis wavefront aberrations that are nearly identical to those of the Escudero-Sanz eye model angle as shown in Fig. 7. The RMS of the difference between corresponding wavefronts is less than 5% of the total wavefront error at each eccentricity. This result demonstrates that a reduced eye can mimic very closely the wavefront aberrations of a much more complex model over a wide field of view. Consequently, the reduced eye becomes an attractive eye model for theoretical and experimental investigations.

Fig. 6. Specification of the 2-surface model eye. R1 and R2: radius of the 1st and 2nd surface; K1 and K2: conic constant of the 1st and 2nd surface; T1 and T2: the distances of the 1st and 2nd surfaces relative to the iris aperture. N: refractive index.
Fig. 7. Comparison of Off-axis wavefront aberrations between the two models (6mm entrance pupil). The wavelength used to calculate the off-axis wavefront aberrations for WAS model is 589 nm.

Although our test cases were limited to modeling the peripheral optical quality along the horizontal meridian, the OWT methodology is easily generalized to include the two dimensional visual field. The OWT method can also be used as a foundation for optimizing the correction of the eye’s optical aberrations by improving the design of contact lenses, interocular lenses, and corneal refractive surgery. Our experience with two test cases described in this report indicates that OWT is a valid concept that can be implemented successfully with the optimization module in Zemax. Future work will be directed at (1) characterizing and developing the optimization algorithms for human eyes from the perspectives of uniqueness, robustness, dimensionality, and achieving a global minimum; (2) incorporating the OWT approach with other auxiliary biometric parameter measuring techniques (e.g. MRI, corneal topography); (3) incorporating this OWT method with other recent proposed optical tomography methods [3

3. J. A. Sakamoto, H. H. Barrett, and A. V. Goncharov, “Inverse optical design of the human eye using likelihood methods and wavefront sensing,” Opt. Express 16, 304–314 (2008). [CrossRef] [PubMed]

5

5. A. S. Goncharov and A. V. Larichev, “Specialized modal tomography of human eye aberrations,” Proc. SPIE 6734, 67341V (2007).

].

Acknowledgments

This research was funded by grant R01-EY05109 from the National Eye Institute of the National Institutes of Health.

References and links

1.

E. Acosta, D. Vazquez, L. Garner, and G. Smith, “Tomographic method for measurement of the gradient refractive index of the crystalline lens. I. The spherical fish lens,” J. Opt. Soc. Am. 22, 424–433 (2005). [CrossRef]

2.

D. Vazquez, E. Acosta, G. Smith, and L. Garner, “Tomographic method for measurement of the gradient refractive index of the crystalline lens. II. The rotationally symmetrical lens,” J. Opt. Soc. Am. 23, 2551–2565 (2006). [CrossRef]

3.

J. A. Sakamoto, H. H. Barrett, and A. V. Goncharov, “Inverse optical design of the human eye using likelihood methods and wavefront sensing,” Opt. Express 16, 304–314 (2008). [CrossRef] [PubMed]

4.

A. Lambert, B. J. Birt, D. A. Atchison, and H. Guo, “Applying SLODAR to measure aberrations in the eye,” Opt. Express 16, 7309–7322 (2008). [CrossRef] [PubMed]

5.

A. S. Goncharov and A. V. Larichev, “Specialized modal tomography of human eye aberrations,” Proc. SPIE 6734, 67341V (2007).

6.

R. Navarro, L. Gonzalez, and J. L. Hernandez-Matamoros, “On the prediction of optical aberrations by personalized eye models,” Optom Vis Sci 83, 371–381 (2006). [CrossRef] [PubMed]

7.

A. V. Goncharov and C. Dainty, “Wide-field schematic eye models with gradient-index lens,” J. Opt. Soc. Am. 24, 2157–2174 (2007). [CrossRef]

8.

A. V. Goncharov, M. Nowakowski, M. T. Sheehan, and C. Dainty, “Reconstruction of the optical system of the human eye with reverse ray-tracing,” Opt. Express 16, 1692–1703 (2008). [CrossRef] [PubMed]

9.

D. A. Atchison, “Optical models for human myopic eyes,” Vision Res. 46, 2236–2250 (2006). [CrossRef] [PubMed]

10.

I. Escudero-Sanz and R. Navarro, “Off-axis aberrations of a wide-angle schematic eye model,” J. Opt. Soc. Am. 16, 1881–1891 (1999). [CrossRef]

11.

H. Sumita, “Orthonormal expansion of the aberration difference function and its application to image evaluation,” Jpn. J. Appl. Phys. 8, 1027–1036 (1969). [CrossRef]

12.

F. Roddier and C. Roddier, “Wavefront reconstruction using iterative Fourier transforms,” Appl. Opt. 30, 1325–1327 (1991). [CrossRef] [PubMed]

13.

D. A. Atchison and D. H. Scott, “Monochromatic aberrations of human eyes in the horizontal visual field,” J. Opt. Soc. Am. 19, 2180–2184 (2002). [CrossRef]

14.

L. Lundstrom, P. Unsbo, and J. Gustafsson, “Off-axis wave front measurements for optical correction in eccentric viewing,” J. Biomed. Opt. 10, 034002 (2005). [CrossRef] [PubMed]

15.

W. Zou and J. P. Rolland, “Iterative zonal wave-front estimation algorithm for optical testing with generalshaped pupils,” J. Opt. Soc. Am. 22, 938–951 (2005). [CrossRef]

16.

D. A. Atchison, D. H. Scott, and W. N. Charman, “Measuring ocular aberrations in the peripheral visual field using Hartmann-Shack aberrometry,” J. Opt. Soc. Am. 24, 2963–2973 (2007). [CrossRef]

17.

Zemax User Guide (Zemax Development Corporation, 2006).

18.

R. Shannon, The Art and Science of Optical Design (Cambridge Univ Press, 1997).

19.

M. Kidger, Intermediate Optical Design (SPIE, 2004). [CrossRef]

20.

G. Smith, D. A. Atchison, C. Avudainayagam, and K. Avudainayagam, “Designing lenses to correct peripheral refractive errors of the eye,” J. Opt. Soc. Am. 19, 10–18 (2002). [CrossRef]

21.

L. N. Thibos, M. Ye, X. Zhang, and A. Bradley, “Spherical aberration of the reduced schematic eye with elliptical refracting surface,” Optom Vision Sci. 74, 548–556 (1997). [CrossRef]

22.

Y. Z. Wang and L. N. Thibos, “Oblique (off-axis) astigmatism of the reduced schematic eye with elliptical refracting surface,” Optom Vis Sci 74, 557–562 (1997). [CrossRef] [PubMed]

OCIS Codes
(330.7325) Vision, color, and visual optics : Visual optics, metrology
(330.7326) Vision, color, and visual optics : Visual optics, modeling

ToC Category:
Vision, Color, and Visual Optics

History
Original Manuscript: October 3, 2008
Revised Manuscript: November 8, 2008
Manuscript Accepted: November 11, 2008
Published: November 25, 2008

Virtual Issues
Vol. 4, Iss. 2 Virtual Journal for Biomedical Optics

Citation
Xin Wei and Larry Thibos, "Modeling the eye's optical system by ocular wavefront tomography," Opt. Express 16, 20490-20502 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-25-20490


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References

  1. E. Acosta, D. Vazquez, L. Garner, and G. Smith, "Tomographic method for measurement of the gradient refractive index of the crystalline lens. I. The spherical fish lens," J. Opt. Soc. Am. 22, 424-433 (2005). [CrossRef]
  2. D. Vazquez, E. Acosta, G. Smith, and L. Garner, "Tomographic method for measurement of the gradient refractive index of the crystalline lens. II. The rotationally symmetrical lens," J. Opt. Soc. Am. 23, 2551-2565 (2006). [CrossRef]
  3. J. A. Sakamoto, H. H. Barrett, and A. V. Goncharov, "Inverse optical design of the human eye using likelihood methods and wavefront sensing," Opt. Express 16, 304-314 (2008). [CrossRef] [PubMed]
  4. A. Lambert, B. J. Birt, D. A. Atchison, and H. Guo, "Applying SLODAR to measure aberrations in the eye," Opt. Express 16, 7309-7322 (2008). [CrossRef] [PubMed]
  5. A. S. Goncharov and A. V. Larichev, "Specialized modal tomography of human eye aberrations," Proc. SPIE 6734, 67341V (2007).
  6. R. Navarro, L. Gonzalez, and J. L. Hernandez-Matamoros, "On the prediction of optical aberrations by personalized eye models," Optom Vis Sci 83, 371-381 (2006). [CrossRef] [PubMed]
  7. A. V. Goncharov and C. Dainty, "Wide-field schematic eye models with gradient-index lens," J. Opt. Soc. Am. 24, 2157-2174 (2007). [CrossRef]
  8. A. V. Goncharov, M. Nowakowski, M. T. Sheehan, and C. Dainty, "Reconstruction of the optical system of the human eye with reverse ray-tracing," Opt. Express 16, 1692-1703 (2008). [CrossRef] [PubMed]
  9. D. A. Atchison, "Optical models for human myopic eyes," Vision Res. 46, 2236-2250 (2006). [CrossRef] [PubMed]
  10. I. Escudero-Sanz and R. Navarro, "Off-axis aberrations of a wide-angle schematic eye model," J. Opt. Soc. Am. 16, 1881-1891 (1999). [CrossRef]
  11. H. Sumita, "Orthonormal expansion of the aberration difference function and its application to image evaluation," Jpn. J. Appl. Phys. 8, 1027-1036 (1969). [CrossRef]
  12. F. Roddier and C. Roddier, "Wavefront reconstruction using iterative Fourier transforms," Appl. Opt. 30, 1325-1327 (1991). [CrossRef] [PubMed]
  13. D. A. Atchison and D. H. Scott, "Monochromatic aberrations of human eyes in the horizontal visual field," J. Opt. Soc. Am. 19, 2180-2184 (2002). [CrossRef]
  14. L. Lundstrom, P. Unsbo, and J. Gustafsson, "Off-axis wave front measurements for optical correction in eccentric viewing," J. Biomed. Opt. 10, 034002 (2005). [CrossRef] [PubMed]
  15. W. Zou and J. P. Rolland, "Iterative zonal wave-front estimation algorithm for optical testing with general-shaped pupils," J. Opt. Soc. Am. 22, 938-951 (2005). [CrossRef]
  16. D. A. Atchison, D. H. Scott, and W. N. Charman, "Measuring ocular aberrations in the peripheral visual field using Hartmann-Shack aberrometry," J. Opt. Soc. Am. 24, 2963-2973 (2007). [CrossRef]
  17. Zemax User Guide (Zemax Development Corporation, 2006).
  18. R. Shannon, The Art and Science of Optical Design (Cambridge Univ Press, 1997).
  19. M. Kidger, Intermediate Optical Design (SPIE, 2004). [CrossRef]
  20. G. Smith, D. A. Atchison, C. Avudainayagam, and K. Avudainayagam, "Designing lenses to correct peripheral refractive errors of the eye," J. Opt. Soc. Am. 19, 10-18 (2002). [CrossRef]
  21. L. N. Thibos, M. Ye, X. Zhang, and A. Bradley, "Spherical aberration of the reduced schematic eye with elliptical refracting surface," Optom Vision Sci. 74, 548-556 (1997). [CrossRef]
  22. Y. Z. Wang and L. N. Thibos, "Oblique (off-axis) astigmatism of the reduced schematic eye with elliptical refracting surface," Optom Vis Sci 74, 557-562 (1997). [CrossRef] [PubMed]

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