Abstract

We are developing a method for estimating patient-specific ocular parameters, including surface curvatures, conic constants, tilts, decentrations, thicknesses, refractive indices, and index gradients. The data consist of the raw detector outputs from one or more Shack-Hartmann wavefront sensors, and the parameters in the eye model are estimated by maximizing the likelihood. A Gaussian noise model is used to emulate electronic noise, so maximum likelihood reduces to nonlinear least-squares fitting between the data and the output of our optical design program. The Fisher information matrix for the Gaussian model was explored to compute bounds on the variance of the estimates for different system configurations. In this preliminary study, an accurate estimate of a chosen subset of ocular parameters was obtained using a custom search algorithm and a nearby starting point to avoid local minima in the complex likelihood surface.

© 2008 Optical Society of America

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References

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  1. J. Tabernero, A. Benito, V. Nourrit, and P. Artal, "Instrument for measuring the misalignments of ocular surfaces," Opt. Express 14, 10945 (2006). http://www.opticsexpress.org/abstract.cfm?id=116620
    [CrossRef] [PubMed]
  2. P. Rosales and S. Marcos, "Phakometry and lens tilt and decentration using a custom-developed Purkinje imaging apparatus: validation and measurements," J. Opt. Soc. Am. A 23, 509 (2006).
    [CrossRef]
  3. P. Rosales, M. Dubbelman, S. Marcos, and G. L. Van der Heijde, "Crystalline lens radii of curvature from Purkinje and Scheimpflug imaging," J. Vis. 6, 1057 (2006).
    [CrossRef] [PubMed]
  4. C. E. Jones, D. A. Atchison, R. Meder, and J. M. Pope, "Refractive index distribution and optical properties of the isolated human lens measured using magnetic resonance imaging (MRI)," Vision Res. 45, 2352 (2005).
    [CrossRef] [PubMed]
  5. B. A. Moffat, D. A. Atchison, and J. M. Pope, "Age-related changes in refractive index distribution and power of the human lens as measured by magnetic resonance micro-imaging in vitro," Vision Res. 42, 1683 (2002).
    [CrossRef] [PubMed]
  6. H. Hofer, L. Chen, G. Yoon, B. Singer, Y. Yamauchi, and D. R. Williams, "Improvement in retinal image quality with dynamic correction of the eye's aberrations," Opt. Express 8, 631 (2001). http://www.opticsexpress.org/abstract.cfm?id=64333
    [CrossRef] [PubMed]
  7. A. Roorda, F. Romero-Borja, W. Donnelly, III, H. Queener, T. Hebert, and M. Campbell, "Adaptive optics scanning laser ophthalmoscopy," Opt. Express 10, 405 (2002). http://www.opticsexpress.org/abstract.cfm?id=68843
    [PubMed]
  8. J. Liang, D. R. Williams, and D. Miller, "Supernormal vision and high-resolution retinal imaging through adaptive optics," J. Opt. Soc. Am. A 14, 2884 (1997).
    [CrossRef]
  9. R. Navarro, J. Santamaría, and J. Bescós, "Accommodation-dependent model of the human eye with aspherics," J. Opt. Soc. Am. A 2, 1273 (1985).
    [CrossRef] [PubMed]
  10. M. Dubbelman and G. L. Van der Heijde, "The shape of the aging human lens: curvature, equivalent refractive index and the lens paradox," Vision Res. 41, 1867 (2001).
    [CrossRef] [PubMed]
  11. R. Navarro, E. Moreno, and C. Dorronsoro, "Monochromatic aberrations and point-spread functions of the human eye across the visual field," J. Opt. Soc. Am. A 15, 2522 (1998).
    [CrossRef]
  12. M. T. Sheehan, A. V. Goncharov, V. M. O’Dwyer, V. Toal, and C. Dainty, "Population study of the variation in monochromatic aberrations of the normal human eye over the central visual field," Opt. Express 15, 7367 (2007).
    [CrossRef] [PubMed]
  13. I. Escudero-Sanz and R. Navarro, "Off-axis aberrations of a wide-angle schematic eye model," J. Opt. Soc. Am. A 16, 1881 (1999).
    [CrossRef]
  14. E. Mallen and P. Kashyap, "Technical note: measurement of retinal contour and supine axial length using the Zeiss IOLMaster," Ophthal. Physiol. Opt. 27, 404 (2007).
    [CrossRef]
  15. P. Artal and A. Guirao, "Contributions of the cornea and the lens to the aberrations of the human eye," Opt. Lett. 23, 1713 (1998).
    [CrossRef]
  16. S. Bará and R. Navarro, "Wide-field compensation of monochromatic eye aberrations: expected performance and design trade-offs," J. Opt. Soc. Am. A 20, 1 (2003).
    [CrossRef]
  17. M. Rynders, B. Lidkea, W. Chisholm, and L. N. Thibos, "Statistical distribution of foveal transverse chromatic aberration, pupil centration, and angle psi in a population of young adult eyes," J. Opt. Soc. Am. A 12, 2348 (1995).
    [CrossRef]
  18. D. Redding, P. Dumont, and J. Yu, "Hubble Space Telescope prescription retrieval," Appl. Opt. 32, 1728 (1993).
    [CrossRef] [PubMed]
  19. G. R. Brady and J. Fienup, "Phase retrieval as an optical metrology tool," in Optifab: Technical digest, SPIE Technical Digest TD03, pp. 139-141 (2005).
  20. J. Schwiegerling, Field Guide to Visual and Ophthalmic Optics (SPIE, 2004).
    [CrossRef]
  21. J. F. Koretz, S. A. Strenk, LawrenceM. Strenk, and John L. Semmlow, "Scheimpflug and high-resolution magnetic resonance imaging of the anterior segment: a comparative study," J. Opt. Soc. Am. A 21, 346 (2004).
    [CrossRef]
  22. M. Dubbelman, H. A. Weeber, G. L. Van der Heijde, and H. J. Völker-Dieben, "Radius and asphericity of the posterior corneal surface determined by corrected Scheimpflug photography," Acta. Ophthalmol. Scand. 80, 379 (2002).
    [CrossRef] [PubMed]
  23. S. A. Strenk, J. L. Semmlow, L. M. Strenk, P. Munoz, J. Gronlund-Jacob, and J. K. DeMarco, "Age-related changes in human ciliary muscle and lens: a magnetic resonance imaging study," Invest. Ophthalmol. Visual Sci. 40, 1162 (1999).
  24. J. Schwiegerling, J. E. Greivenkamp, and J. M. Miller, "Representation of videokeratoscopic height data with Zernike polynomials," J. Opt. Soc. Am. A 12, 2105 (1995).
    [CrossRef]
  25. R. Navarro, L. González, and J. L. Hernández, "Optics of the average normal cornea from general and canonical representations of its surface topography," J. Opt. Soc. Am. A 23, 219 (2006).
    [CrossRef]
  26. F. Zhou, X. Hong, D. T. Miller, L. N. Thibos, and A. Bradley, "Validation of a combined corneal topographer and aberrometer based on Shack-Hartmann wave-front sensing," J. Opt. Soc. Am. A 21, 683 (2004).
    [CrossRef]
  27. A. Guirao and P. Artal, "Corneal wave aberration from videokeratography: accuracy and limitations of the procedure," J. Opt. Soc. Am. A 17, 955 (2000).
    [CrossRef]
  28. J. Straub, J. Schwiegerling, and A. Gupta, "Design of a compact Shack-Hartmann aberrometer for real-time measurement of aberrations in human eyes" in Vision Science and Its Applications, OSA Technical Digest (Optical Society of America, Washington DC, 2001), pp. 110-113.
  29. D. A. Atchison and G. Smith, "Chromatic dispersions of the ocular media of human eyes," J. Opt. Soc. Am. A 22, 29 (2005).
    [CrossRef]
  30. A. V. Goncharov and C. Dainty, "Wide-field schematic eye models with gradient-index lens," J. Opt. Soc. Am. A 24, 2157 (2007).
    [CrossRef]
  31. R. Navarro, F. Palos, and L. M. González, "Adaptive model of the gradient index of the human lens. I. optics formulation and model of aging ex vivo lenses," J. Opt. Soc. Am. A 24, 2175 (2007).
    [CrossRef]
  32. R. Navarro, F. Palos, and L. M. González, "Adaptive model of the gradient index of the human lens. II. optics of the accommodating aging lens," J. Opt. Soc. Am. A 24, 2911 (2007).
    [CrossRef]
  33. H. H. Barrett, C. Dainty, and D. Lara, "Maximum-likelihood methods in wavefront sensing: stochastic models and likelihood functions," J. Opt. Soc. Am. A 24, 391 (2007).
    [CrossRef]
  34. 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 (to be published).

2007

2006

2005

C. E. Jones, D. A. Atchison, R. Meder, and J. M. Pope, "Refractive index distribution and optical properties of the isolated human lens measured using magnetic resonance imaging (MRI)," Vision Res. 45, 2352 (2005).
[CrossRef] [PubMed]

D. A. Atchison and G. Smith, "Chromatic dispersions of the ocular media of human eyes," J. Opt. Soc. Am. A 22, 29 (2005).
[CrossRef]

2004

2003

2002

A. Roorda, F. Romero-Borja, W. Donnelly, III, H. Queener, T. Hebert, and M. Campbell, "Adaptive optics scanning laser ophthalmoscopy," Opt. Express 10, 405 (2002). http://www.opticsexpress.org/abstract.cfm?id=68843
[PubMed]

M. Dubbelman, H. A. Weeber, G. L. Van der Heijde, and H. J. Völker-Dieben, "Radius and asphericity of the posterior corneal surface determined by corrected Scheimpflug photography," Acta. Ophthalmol. Scand. 80, 379 (2002).
[CrossRef] [PubMed]

B. A. Moffat, D. A. Atchison, and J. M. Pope, "Age-related changes in refractive index distribution and power of the human lens as measured by magnetic resonance micro-imaging in vitro," Vision Res. 42, 1683 (2002).
[CrossRef] [PubMed]

2001

2000

1999

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

S. A. Strenk, J. L. Semmlow, L. M. Strenk, P. Munoz, J. Gronlund-Jacob, and J. K. DeMarco, "Age-related changes in human ciliary muscle and lens: a magnetic resonance imaging study," Invest. Ophthalmol. Visual Sci. 40, 1162 (1999).

1998

1997

1995

1993

1985

Acta. Ophthalmol. Scand.

M. Dubbelman, H. A. Weeber, G. L. Van der Heijde, and H. J. Völker-Dieben, "Radius and asphericity of the posterior corneal surface determined by corrected Scheimpflug photography," Acta. Ophthalmol. Scand. 80, 379 (2002).
[CrossRef] [PubMed]

Appl. Opt.

Invest. Ophthalmol. Visual Sci.

S. A. Strenk, J. L. Semmlow, L. M. Strenk, P. Munoz, J. Gronlund-Jacob, and J. K. DeMarco, "Age-related changes in human ciliary muscle and lens: a magnetic resonance imaging study," Invest. Ophthalmol. Visual Sci. 40, 1162 (1999).

J. Opt. Soc. Am. A

R. Navarro, J. Santamaría, and J. Bescós, "Accommodation-dependent model of the human eye with aspherics," J. Opt. Soc. Am. A 2, 1273 (1985).
[CrossRef] [PubMed]

J. Schwiegerling, J. E. Greivenkamp, and J. M. Miller, "Representation of videokeratoscopic height data with Zernike polynomials," J. Opt. Soc. Am. A 12, 2105 (1995).
[CrossRef]

A. Guirao and P. Artal, "Corneal wave aberration from videokeratography: accuracy and limitations of the procedure," J. Opt. Soc. Am. A 17, 955 (2000).
[CrossRef]

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

R. Navarro, E. Moreno, and C. Dorronsoro, "Monochromatic aberrations and point-spread functions of the human eye across the visual field," J. Opt. Soc. Am. A 15, 2522 (1998).
[CrossRef]

J. Liang, D. R. Williams, and D. Miller, "Supernormal vision and high-resolution retinal imaging through adaptive optics," J. Opt. Soc. Am. A 14, 2884 (1997).
[CrossRef]

M. Rynders, B. Lidkea, W. Chisholm, and L. N. Thibos, "Statistical distribution of foveal transverse chromatic aberration, pupil centration, and angle psi in a population of young adult eyes," J. Opt. Soc. Am. A 12, 2348 (1995).
[CrossRef]

S. Bará and R. Navarro, "Wide-field compensation of monochromatic eye aberrations: expected performance and design trade-offs," J. Opt. Soc. Am. A 20, 1 (2003).
[CrossRef]

J. F. Koretz, S. A. Strenk, LawrenceM. Strenk, and John L. Semmlow, "Scheimpflug and high-resolution magnetic resonance imaging of the anterior segment: a comparative study," J. Opt. Soc. Am. A 21, 346 (2004).
[CrossRef]

F. Zhou, X. Hong, D. T. Miller, L. N. Thibos, and A. Bradley, "Validation of a combined corneal topographer and aberrometer based on Shack-Hartmann wave-front sensing," J. Opt. Soc. Am. A 21, 683 (2004).
[CrossRef]

D. A. Atchison and G. Smith, "Chromatic dispersions of the ocular media of human eyes," J. Opt. Soc. Am. A 22, 29 (2005).
[CrossRef]

R. Navarro, L. González, and J. L. Hernández, "Optics of the average normal cornea from general and canonical representations of its surface topography," J. Opt. Soc. Am. A 23, 219 (2006).
[CrossRef]

P. Rosales and S. Marcos, "Phakometry and lens tilt and decentration using a custom-developed Purkinje imaging apparatus: validation and measurements," J. Opt. Soc. Am. A 23, 509 (2006).
[CrossRef]

H. H. Barrett, C. Dainty, and D. Lara, "Maximum-likelihood methods in wavefront sensing: stochastic models and likelihood functions," J. Opt. Soc. Am. A 24, 391 (2007).
[CrossRef]

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

R. Navarro, F. Palos, and L. M. González, "Adaptive model of the gradient index of the human lens. I. optics formulation and model of aging ex vivo lenses," J. Opt. Soc. Am. A 24, 2175 (2007).
[CrossRef]

R. Navarro, F. Palos, and L. M. González, "Adaptive model of the gradient index of the human lens. II. optics of the accommodating aging lens," J. Opt. Soc. Am. A 24, 2911 (2007).
[CrossRef]

J. Vis.

P. Rosales, M. Dubbelman, S. Marcos, and G. L. Van der Heijde, "Crystalline lens radii of curvature from Purkinje and Scheimpflug imaging," J. Vis. 6, 1057 (2006).
[CrossRef] [PubMed]

Ophthal. Physiol. Opt.

E. Mallen and P. Kashyap, "Technical note: measurement of retinal contour and supine axial length using the Zeiss IOLMaster," Ophthal. Physiol. Opt. 27, 404 (2007).
[CrossRef]

Opt. Express

Opt. Lett.

Vision Res.

C. E. Jones, D. A. Atchison, R. Meder, and J. M. Pope, "Refractive index distribution and optical properties of the isolated human lens measured using magnetic resonance imaging (MRI)," Vision Res. 45, 2352 (2005).
[CrossRef] [PubMed]

B. A. Moffat, D. A. Atchison, and J. M. Pope, "Age-related changes in refractive index distribution and power of the human lens as measured by magnetic resonance micro-imaging in vitro," Vision Res. 42, 1683 (2002).
[CrossRef] [PubMed]

M. Dubbelman and G. L. Van der Heijde, "The shape of the aging human lens: curvature, equivalent refractive index and the lens paradox," Vision Res. 41, 1867 (2001).
[CrossRef] [PubMed]

Other

G. R. Brady and J. Fienup, "Phase retrieval as an optical metrology tool," in Optifab: Technical digest, SPIE Technical Digest TD03, pp. 139-141 (2005).

J. Schwiegerling, Field Guide to Visual and Ophthalmic Optics (SPIE, 2004).
[CrossRef]

J. Straub, J. Schwiegerling, and A. Gupta, "Design of a compact Shack-Hartmann aberrometer for real-time measurement of aberrations in human eyes" in Vision Science and Its Applications, OSA Technical Digest (Optical Society of America, Washington DC, 2001), pp. 110-113.

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Figures (6)

Fig. 1.
Fig. 1.

Shack-Hartmann WFS system for the eye. In our computational model, we exclude relay optics and place the center of the lenslet array 10 mm from the corneal apex. The angle between the laser beam and the optical axis of the eye is denoted α.

Fig. 2.
Fig. 2.

Geometrical eye model used to generate WFS data, corresponding to ocular parameters in Table 1. Sample rays for 0, 6, and 12 degrees are shown. The y-axis corresponds to the vertical direction.

Fig. 3.
Fig. 3.

Corresponding WFS data used as input to inverse optical design, for beam angle: (a) 0 degrees, (b) 6 degrees, and (c) 12 degrees.

Fig. 4.
Fig. 4.

Focal spots on the retina due to the source beam for angle: (a) 0 degrees, (b) 6 degrees, and (c) 12 degrees.

Fig. 5.
Fig. 5.

(a) FIM for the system of beam angles on a logarithmic scale. (b) Inverse of the FIM, indicating coupling between parameter estimates and providing the CRBs as the diagonal entries.

Fig. 6.
Fig. 6.

Reconstructed eye model of the estimated parameters, superimposed with the true values in the data.

Tables (2)

Tables Icon

Table 1. Geometry of Eye Model Used in Simulation at 780 nm

Tables Icon

Table 2. Estimated ocular parameters, including the values used to generate the data, standard deviations computed from the lower CRBs, starting point in the ML search, parameter estimates, and fractional error in the estimates.

Equations (7)

Equations on this page are rendered with MathJax. Learn more.

pr ( g | θ ) = Π m = 1 M 1 2 π σ 2 exp [ [ g m g ¯ m ( θ ) ] 2 2 σ 2 ] ,
θ ̂ ML arg max θ pr ( g | θ ) ,
θ ̂ ML arg max θ ln [ pr ( g | θ ) ] .
θ ̂ ML = arg min θ m = 1 M [ g m g ¯ m ( θ ) ] 2 .
F j k = [ θ j l n p r ( g θ ) ] [ θ k l n p r ( g θ ) ] g θ ,
F j k = 1 σ 2 m = 1 M g ¯ m ( θ ) θ j g ¯ m ( θ ) θ k ,
[ K θ ̂ ] nn = V a r { θ ̂ n } [ F 1 ] n n .

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