Abstract

In order to work in a consistent way with Zernike aberration coefficients estimated in different pupils, it is necessary to refer them to a common pupil size. Two standard approaches can be used to that end: to rescale algebraically the coefficients estimated in the original pupil or to refit them anew using the wavefront slope measurements available within the new one. These procedures are not equivalent; they are affected by different estimation errors that we address in this work. Our results for normal eye populations show that in case of reducing the pupil size it is better to rescale the original coefficients than to refit them using the measurements contained within the smaller pupil. In case of enlarging the pupil size, as it can a priori be expected, the opposite holds true. We provide explicit expressions to quantify the errors arising in both cases, including the expected error incurred when extrapolating the Zernike estimation beyond the radius where the measurements were made.

© 2013 Optical Society of America

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References

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  1. M. Born and E. Wolf, Principles of Optics (Cambridge University, 1998), pp. 464–466, 767–772.
  2. L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, and R. Webb, VSIA Standards Taskforce Members, “Standards for reporting the optical aberrations of eyes,” in Vision Science and Its Applications, V. Lakshminarayanan, ed., Vol. 35 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2000), pp. 232–244.
  3. T. O. Salmon and C. van de Pol, “Normal-eye Zernike coefficients and root-mean-square wavefront errors,” J. Cataract Refractive Surg. 32, 2064–2074 (2006).
    [CrossRef]
  4. E. Moreno-Barriuso, J. Merayo-Lloves, S. Marcos, R. Navarro, L. Llorente, and S. Barbero, “Ocular aberrations before and after myopic corneal refractive surgery: LASIK-induced changes measured with laser ray tracing,” Investig. Ophthalmol. Visual Sci. 42, 1396–1403 (2001).
  5. R. Montés-Micó, P. Hernández, V. Fernández-Sánchez, S. Bonaque, F. Lara, and N. López-Gil, “Changes of the eye optics after iris constriction,” J. Optom. 3, 212–218 (2010).
  6. L. N. Thibos, A. Bradley, and N. Lopez-Gil, “Modelling the impact of spherical aberration on accommodation,” Ophthalmic Physiolog. Opt. 33, 482–496 (2013).
    [CrossRef]
  7. S. Bará, E. Pailos, and J. Arines, “Signal-to-noise ratio and aberration statistics in ocular aberrometry,” Opt. Lett. 37, 2427–2429 (2012).
    [CrossRef]
  8. K. A. Goldberg and K. Geary, “Wave-front measurement errors from restricted concentric subdomains,” J. Opt. Soc. Am. A 18, 2146–2152 (2001).
    [CrossRef]
  9. J. Schwiegerling, “Scaling Zernike expansion coefficients to different pupil sizes,” J. Opt. Soc. Am. A 19, 1937–1945 (2002).
    [CrossRef]
  10. C. E. Campbell, “Matrix method to find a new set of Zernike coefficients from an original set when the aperture radius is changed,” J. Opt. Soc. Am. A 20, 209–217 (2003).
    [CrossRef]
  11. G. M. Dai, “Scaling Zernike expansion coefficients to smaller pupil sizes: a simpler formula,” J. Opt. Soc. Am. A 23, 539–543 (2006).
    [CrossRef]
  12. H. Shu, L. Luo, G. Han, and J. L. Coatrieux, “General method to derive the relationship between two sets of Zernike coefficients corresponding to different aperture sizes,” J. Opt. Soc. Am. A 23, 1960–1968 (2006).
    [CrossRef]
  13. S. Bará, J. Arines, J. Ares, and P. Prado, “Direct transformation of Zernike eye aberration coefficients between scaled, rotated and/or displaced pupils,” J. Opt. Soc. Am. A 23, 2061–2066 (2006).
    [CrossRef]
  14. A. J. Janssen and P. Dirksen, “Concise formula for the Zernike coefficients of scaled pupils,” J. Microlithogr. Microfabr. Microsyst. 5, 030501 (2006).
    [CrossRef]
  15. L. Lundström and P. Unsbo, “Transformation of Zernike coefficients: scaled, translated, and rotated wavefronts with circular and elliptical pupils,” J. Opt. Soc. Am. A 24, 569–577 (2007).
    [CrossRef]
  16. A. J. Janssen, S. van Haver, P. Dirksen, and J. J. Braat, “Zernike representation and Strehl ratio of optical systems with variable numerical aperture,” J. Mod. Opt. 55, 1127–1157 (2008).
    [CrossRef]
  17. J. A. Díaz, J. Fernández-Dorado, C. Pizarro, and J. Arasa, “Zernike coefficients for concentric, circular scaled pupils: an equivalent expression,” J. Mod. Opt. 56, 131–137 (2009).
    [CrossRef]
  18. K. Dillon, “Bilinear wavefront transformation,” J. Opt. Soc. Am. A 26, 1839–1846 (2009).
    [CrossRef]
  19. V. N. Mahajan, “Zernike coefficients of a scaled pupil,” Appl. Opt. 49, 5374–5377 (2010).
    [CrossRef]
  20. E. Tatulli, “Transformation of Zernike coefficients: a Fourier-based method for scaled, translated, and rotated wavefront apertures,” J. Opt. Soc. Am. A 30, 726–732 (2013).
    [CrossRef]
  21. J. Arines, P. Prado, S. Bará, and E. Acosta, “Equivalence of least-squares estimation of eye aberrations in linearly transformed reference frames,” Opt. Commun. 281, 2716–2721 (2008).
    [CrossRef]
  22. S. Bará, P. Prado, J. Arines, and J. Ares, “Estimation-induced correlations of the Zernike coefficients of the eye aberration,” Opt. Lett. 31, 2646–2648 (2006).
    [CrossRef]
  23. J. Porter, A. Guirao, I. G. Cox, and D. R. Williams, “Monochromatic aberrations of the human eye in a large population,” J. Opt. Soc. Am. A 18, 1793–1803 (2001).
    [CrossRef]
  24. L. N. Thibos, X. Hong, A. Bradley, and X. Cheng, “Statistical variation of aberration structure and image quality in a normal population of healthy eyes,” J. Opt. Soc. Am. A 19, 2329–2348 (2002).
    [CrossRef]
  25. J. F. Castejon-Mochon, N. Lopez-Gil, A. Benito, and P. Artal, “Ocular wave-front aberration statistics in a normal young population,” Vis. Res. 42, 1611–1617 (2002).
    [CrossRef]
  26. T. R. Candy and J. Wang, “Higher order monochromatic aberrations of the human infant eye,” J. Vis. 5(12):6, 543–545 (2005).
    [CrossRef]
  27. C. Leahy, C. Leroux, C. Dainty, and L. Diaz-Santana, “Temporal dynamics and statistical characteristics of the microfluctuations of accommodation: dependence on the mean accommodative effort,” Opt. Express 18, 2668–2681 (2010).
    [CrossRef]
  28. E. Pailos, A. Ommani, L. Díaz-Santana, and S. Bará, “Centroid displacement statistics of the eye aberration,” J. Opt. Soc. Am. A 27, 1818–1827 (2010).
    [CrossRef]
  29. J. Liang, W. Grimm, S. Goelz, and J. F. Bille, “Objective measurement of the wave aberrations of the human eye using a Hartmann–Shack wave-front sensor,” J. Opt. Soc. Am. A 11, 1949–1957 (1994).
    [CrossRef]
  30. J. Liang and D. R. Williams, “Aberrations and retinal image quality of the normal human eye,” J. Opt. Soc. Am. A 14, 2873–2883 (1997).
    [CrossRef]
  31. R. Navarro and M. A. Losada, “Aberrations and relative efficiency of light pencils in the living human eye,” Optom. Vis. Sci. 74, 540–547 (1997).
    [CrossRef]
  32. R. Navarro and E. Moreno-Barriuso, “Laser ray-tracing method for optical testing,” Opt. Lett. 24, 951–953 (1999).
    [CrossRef]
  33. R. H. Webb, C. M. Penney, and K. P. Thompson, “Measurement of ocular wavefront distortion with a spatially resolved refractometer,” Appl. Opt. 31, 3678–3686 (1992).
    [CrossRef]
  34. J. C. He, S. Marcos, R. H. Webb, and S. A. Burns, “Measurement of the wavefront aberration of the eye by a fast psychophysical procedure,” J. Opt. Soc. Am. A 15, 2449–2456 (1998).
    [CrossRef]
  35. E. P. Wallner, “Optimal wave-front correction using slope measurements,” J. Opt. Soc. Am. 73, 1771–1776 (1983).
    [CrossRef]
  36. L. Díaz-Santana, G. Walker, and S. X. Bará, “Sampling geometries for ocular aberrometry: a model for evaluation of performance,” Opt. Express 13, 8801–8818 (2005).
    [CrossRef]
  37. O. Soloviev and G. Vdovin, “Hartmann–Shack test with random masks for modal wavefront reconstruction,” Opt. Express 13, 9570–9584 (2005).
    [CrossRef]
  38. S. Bará, “Measuring eye aberrations with Hartmann–Shack wave-front sensors: should the irradiance distribution across the eye pupil be taken into account?” J. Opt. Soc. Am. A 20, 2237–2245 (2003).
    [CrossRef]
  39. J. Herrmann, “Cross coupling and aliasing in modal wave-front estimation,” J. Opt. Soc. Am. 71, 989–992 (1981).
    [CrossRef]
  40. V. I. Tatarskii, The Propagation of Waves in the Turbulent Atmosphere (Nauka, 1967), pp. 385–390 (in Russian).
  41. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976).
    [CrossRef]
  42. J. Arines, E. Pailos, P. Prado, and S. Bará, “The contribution of the fixational eye movements to the variability of the measured ocular aberration,” Ophthalmic Physiolog. Opt. 29, 281–287 (2009).
    [CrossRef]
  43. G. M. Dai, “Validity of scaling Zernike coefficients to a larger diameter for refractive surgery,” J. Refract. Sur. 27, 837–841 (2011).
    [CrossRef]

2013 (2)

L. N. Thibos, A. Bradley, and N. Lopez-Gil, “Modelling the impact of spherical aberration on accommodation,” Ophthalmic Physiolog. Opt. 33, 482–496 (2013).
[CrossRef]

E. Tatulli, “Transformation of Zernike coefficients: a Fourier-based method for scaled, translated, and rotated wavefront apertures,” J. Opt. Soc. Am. A 30, 726–732 (2013).
[CrossRef]

2012 (1)

2011 (1)

G. M. Dai, “Validity of scaling Zernike coefficients to a larger diameter for refractive surgery,” J. Refract. Sur. 27, 837–841 (2011).
[CrossRef]

2010 (4)

2009 (3)

J. A. Díaz, J. Fernández-Dorado, C. Pizarro, and J. Arasa, “Zernike coefficients for concentric, circular scaled pupils: an equivalent expression,” J. Mod. Opt. 56, 131–137 (2009).
[CrossRef]

K. Dillon, “Bilinear wavefront transformation,” J. Opt. Soc. Am. A 26, 1839–1846 (2009).
[CrossRef]

J. Arines, E. Pailos, P. Prado, and S. Bará, “The contribution of the fixational eye movements to the variability of the measured ocular aberration,” Ophthalmic Physiolog. Opt. 29, 281–287 (2009).
[CrossRef]

2008 (2)

A. J. Janssen, S. van Haver, P. Dirksen, and J. J. Braat, “Zernike representation and Strehl ratio of optical systems with variable numerical aperture,” J. Mod. Opt. 55, 1127–1157 (2008).
[CrossRef]

J. Arines, P. Prado, S. Bará, and E. Acosta, “Equivalence of least-squares estimation of eye aberrations in linearly transformed reference frames,” Opt. Commun. 281, 2716–2721 (2008).
[CrossRef]

2007 (1)

2006 (6)

2005 (3)

2003 (2)

2002 (3)

2001 (3)

J. Porter, A. Guirao, I. G. Cox, and D. R. Williams, “Monochromatic aberrations of the human eye in a large population,” J. Opt. Soc. Am. A 18, 1793–1803 (2001).
[CrossRef]

E. Moreno-Barriuso, J. Merayo-Lloves, S. Marcos, R. Navarro, L. Llorente, and S. Barbero, “Ocular aberrations before and after myopic corneal refractive surgery: LASIK-induced changes measured with laser ray tracing,” Investig. Ophthalmol. Visual Sci. 42, 1396–1403 (2001).

K. A. Goldberg and K. Geary, “Wave-front measurement errors from restricted concentric subdomains,” J. Opt. Soc. Am. A 18, 2146–2152 (2001).
[CrossRef]

1999 (1)

1998 (1)

1997 (2)

J. Liang and D. R. Williams, “Aberrations and retinal image quality of the normal human eye,” J. Opt. Soc. Am. A 14, 2873–2883 (1997).
[CrossRef]

R. Navarro and M. A. Losada, “Aberrations and relative efficiency of light pencils in the living human eye,” Optom. Vis. Sci. 74, 540–547 (1997).
[CrossRef]

1994 (1)

1992 (1)

1983 (1)

1981 (1)

1976 (1)

Acosta, E.

J. Arines, P. Prado, S. Bará, and E. Acosta, “Equivalence of least-squares estimation of eye aberrations in linearly transformed reference frames,” Opt. Commun. 281, 2716–2721 (2008).
[CrossRef]

Applegate, R. A.

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, and R. Webb, VSIA Standards Taskforce Members, “Standards for reporting the optical aberrations of eyes,” in Vision Science and Its Applications, V. Lakshminarayanan, ed., Vol. 35 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2000), pp. 232–244.

Arasa, J.

J. A. Díaz, J. Fernández-Dorado, C. Pizarro, and J. Arasa, “Zernike coefficients for concentric, circular scaled pupils: an equivalent expression,” J. Mod. Opt. 56, 131–137 (2009).
[CrossRef]

Ares, J.

Arines, J.

S. Bará, E. Pailos, and J. Arines, “Signal-to-noise ratio and aberration statistics in ocular aberrometry,” Opt. Lett. 37, 2427–2429 (2012).
[CrossRef]

J. Arines, E. Pailos, P. Prado, and S. Bará, “The contribution of the fixational eye movements to the variability of the measured ocular aberration,” Ophthalmic Physiolog. Opt. 29, 281–287 (2009).
[CrossRef]

J. Arines, P. Prado, S. Bará, and E. Acosta, “Equivalence of least-squares estimation of eye aberrations in linearly transformed reference frames,” Opt. Commun. 281, 2716–2721 (2008).
[CrossRef]

S. Bará, P. Prado, J. Arines, and J. Ares, “Estimation-induced correlations of the Zernike coefficients of the eye aberration,” Opt. Lett. 31, 2646–2648 (2006).
[CrossRef]

S. Bará, J. Arines, J. Ares, and P. Prado, “Direct transformation of Zernike eye aberration coefficients between scaled, rotated and/or displaced pupils,” J. Opt. Soc. Am. A 23, 2061–2066 (2006).
[CrossRef]

Artal, P.

J. F. Castejon-Mochon, N. Lopez-Gil, A. Benito, and P. Artal, “Ocular wave-front aberration statistics in a normal young population,” Vis. Res. 42, 1611–1617 (2002).
[CrossRef]

Bará, S.

Bará, S. X.

Barbero, S.

E. Moreno-Barriuso, J. Merayo-Lloves, S. Marcos, R. Navarro, L. Llorente, and S. Barbero, “Ocular aberrations before and after myopic corneal refractive surgery: LASIK-induced changes measured with laser ray tracing,” Investig. Ophthalmol. Visual Sci. 42, 1396–1403 (2001).

Benito, A.

J. F. Castejon-Mochon, N. Lopez-Gil, A. Benito, and P. Artal, “Ocular wave-front aberration statistics in a normal young population,” Vis. Res. 42, 1611–1617 (2002).
[CrossRef]

Bille, J. F.

Bonaque, S.

R. Montés-Micó, P. Hernández, V. Fernández-Sánchez, S. Bonaque, F. Lara, and N. López-Gil, “Changes of the eye optics after iris constriction,” J. Optom. 3, 212–218 (2010).

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1998), pp. 464–466, 767–772.

Braat, J. J.

A. J. Janssen, S. van Haver, P. Dirksen, and J. J. Braat, “Zernike representation and Strehl ratio of optical systems with variable numerical aperture,” J. Mod. Opt. 55, 1127–1157 (2008).
[CrossRef]

Bradley, A.

L. N. Thibos, A. Bradley, and N. Lopez-Gil, “Modelling the impact of spherical aberration on accommodation,” Ophthalmic Physiolog. Opt. 33, 482–496 (2013).
[CrossRef]

L. N. Thibos, X. Hong, A. Bradley, and X. Cheng, “Statistical variation of aberration structure and image quality in a normal population of healthy eyes,” J. Opt. Soc. Am. A 19, 2329–2348 (2002).
[CrossRef]

Burns, S. A.

Campbell, C. E.

Candy, T. R.

T. R. Candy and J. Wang, “Higher order monochromatic aberrations of the human infant eye,” J. Vis. 5(12):6, 543–545 (2005).
[CrossRef]

Castejon-Mochon, J. F.

J. F. Castejon-Mochon, N. Lopez-Gil, A. Benito, and P. Artal, “Ocular wave-front aberration statistics in a normal young population,” Vis. Res. 42, 1611–1617 (2002).
[CrossRef]

Cheng, X.

Coatrieux, J. L.

Cox, I. G.

Dai, G. M.

G. M. Dai, “Validity of scaling Zernike coefficients to a larger diameter for refractive surgery,” J. Refract. Sur. 27, 837–841 (2011).
[CrossRef]

G. M. Dai, “Scaling Zernike expansion coefficients to smaller pupil sizes: a simpler formula,” J. Opt. Soc. Am. A 23, 539–543 (2006).
[CrossRef]

Dainty, C.

Díaz, J. A.

J. A. Díaz, J. Fernández-Dorado, C. Pizarro, and J. Arasa, “Zernike coefficients for concentric, circular scaled pupils: an equivalent expression,” J. Mod. Opt. 56, 131–137 (2009).
[CrossRef]

Diaz-Santana, L.

Díaz-Santana, L.

Dillon, K.

Dirksen, P.

A. J. Janssen, S. van Haver, P. Dirksen, and J. J. Braat, “Zernike representation and Strehl ratio of optical systems with variable numerical aperture,” J. Mod. Opt. 55, 1127–1157 (2008).
[CrossRef]

A. J. Janssen and P. Dirksen, “Concise formula for the Zernike coefficients of scaled pupils,” J. Microlithogr. Microfabr. Microsyst. 5, 030501 (2006).
[CrossRef]

Fernández-Dorado, J.

J. A. Díaz, J. Fernández-Dorado, C. Pizarro, and J. Arasa, “Zernike coefficients for concentric, circular scaled pupils: an equivalent expression,” J. Mod. Opt. 56, 131–137 (2009).
[CrossRef]

Fernández-Sánchez, V.

R. Montés-Micó, P. Hernández, V. Fernández-Sánchez, S. Bonaque, F. Lara, and N. López-Gil, “Changes of the eye optics after iris constriction,” J. Optom. 3, 212–218 (2010).

Geary, K.

Goelz, S.

Goldberg, K. A.

Grimm, W.

Guirao, A.

Han, G.

He, J. C.

Hernández, P.

R. Montés-Micó, P. Hernández, V. Fernández-Sánchez, S. Bonaque, F. Lara, and N. López-Gil, “Changes of the eye optics after iris constriction,” J. Optom. 3, 212–218 (2010).

Herrmann, J.

Hong, X.

Janssen, A. J.

A. J. Janssen, S. van Haver, P. Dirksen, and J. J. Braat, “Zernike representation and Strehl ratio of optical systems with variable numerical aperture,” J. Mod. Opt. 55, 1127–1157 (2008).
[CrossRef]

A. J. Janssen and P. Dirksen, “Concise formula for the Zernike coefficients of scaled pupils,” J. Microlithogr. Microfabr. Microsyst. 5, 030501 (2006).
[CrossRef]

Lara, F.

R. Montés-Micó, P. Hernández, V. Fernández-Sánchez, S. Bonaque, F. Lara, and N. López-Gil, “Changes of the eye optics after iris constriction,” J. Optom. 3, 212–218 (2010).

Leahy, C.

Leroux, C.

Liang, J.

Llorente, L.

E. Moreno-Barriuso, J. Merayo-Lloves, S. Marcos, R. Navarro, L. Llorente, and S. Barbero, “Ocular aberrations before and after myopic corneal refractive surgery: LASIK-induced changes measured with laser ray tracing,” Investig. Ophthalmol. Visual Sci. 42, 1396–1403 (2001).

Lopez-Gil, N.

L. N. Thibos, A. Bradley, and N. Lopez-Gil, “Modelling the impact of spherical aberration on accommodation,” Ophthalmic Physiolog. Opt. 33, 482–496 (2013).
[CrossRef]

J. F. Castejon-Mochon, N. Lopez-Gil, A. Benito, and P. Artal, “Ocular wave-front aberration statistics in a normal young population,” Vis. Res. 42, 1611–1617 (2002).
[CrossRef]

López-Gil, N.

R. Montés-Micó, P. Hernández, V. Fernández-Sánchez, S. Bonaque, F. Lara, and N. López-Gil, “Changes of the eye optics after iris constriction,” J. Optom. 3, 212–218 (2010).

Losada, M. A.

R. Navarro and M. A. Losada, “Aberrations and relative efficiency of light pencils in the living human eye,” Optom. Vis. Sci. 74, 540–547 (1997).
[CrossRef]

Lundström, L.

Luo, L.

Mahajan, V. N.

Marcos, S.

E. Moreno-Barriuso, J. Merayo-Lloves, S. Marcos, R. Navarro, L. Llorente, and S. Barbero, “Ocular aberrations before and after myopic corneal refractive surgery: LASIK-induced changes measured with laser ray tracing,” Investig. Ophthalmol. Visual Sci. 42, 1396–1403 (2001).

J. C. He, S. Marcos, R. H. Webb, and S. A. Burns, “Measurement of the wavefront aberration of the eye by a fast psychophysical procedure,” J. Opt. Soc. Am. A 15, 2449–2456 (1998).
[CrossRef]

Merayo-Lloves, J.

E. Moreno-Barriuso, J. Merayo-Lloves, S. Marcos, R. Navarro, L. Llorente, and S. Barbero, “Ocular aberrations before and after myopic corneal refractive surgery: LASIK-induced changes measured with laser ray tracing,” Investig. Ophthalmol. Visual Sci. 42, 1396–1403 (2001).

Montés-Micó, R.

R. Montés-Micó, P. Hernández, V. Fernández-Sánchez, S. Bonaque, F. Lara, and N. López-Gil, “Changes of the eye optics after iris constriction,” J. Optom. 3, 212–218 (2010).

Moreno-Barriuso, E.

E. Moreno-Barriuso, J. Merayo-Lloves, S. Marcos, R. Navarro, L. Llorente, and S. Barbero, “Ocular aberrations before and after myopic corneal refractive surgery: LASIK-induced changes measured with laser ray tracing,” Investig. Ophthalmol. Visual Sci. 42, 1396–1403 (2001).

R. Navarro and E. Moreno-Barriuso, “Laser ray-tracing method for optical testing,” Opt. Lett. 24, 951–953 (1999).
[CrossRef]

Navarro, R.

E. Moreno-Barriuso, J. Merayo-Lloves, S. Marcos, R. Navarro, L. Llorente, and S. Barbero, “Ocular aberrations before and after myopic corneal refractive surgery: LASIK-induced changes measured with laser ray tracing,” Investig. Ophthalmol. Visual Sci. 42, 1396–1403 (2001).

R. Navarro and E. Moreno-Barriuso, “Laser ray-tracing method for optical testing,” Opt. Lett. 24, 951–953 (1999).
[CrossRef]

R. Navarro and M. A. Losada, “Aberrations and relative efficiency of light pencils in the living human eye,” Optom. Vis. Sci. 74, 540–547 (1997).
[CrossRef]

Noll, R. J.

Ommani, A.

Pailos, E.

Penney, C. M.

Pizarro, C.

J. A. Díaz, J. Fernández-Dorado, C. Pizarro, and J. Arasa, “Zernike coefficients for concentric, circular scaled pupils: an equivalent expression,” J. Mod. Opt. 56, 131–137 (2009).
[CrossRef]

Porter, J.

Prado, P.

J. Arines, E. Pailos, P. Prado, and S. Bará, “The contribution of the fixational eye movements to the variability of the measured ocular aberration,” Ophthalmic Physiolog. Opt. 29, 281–287 (2009).
[CrossRef]

J. Arines, P. Prado, S. Bará, and E. Acosta, “Equivalence of least-squares estimation of eye aberrations in linearly transformed reference frames,” Opt. Commun. 281, 2716–2721 (2008).
[CrossRef]

S. Bará, P. Prado, J. Arines, and J. Ares, “Estimation-induced correlations of the Zernike coefficients of the eye aberration,” Opt. Lett. 31, 2646–2648 (2006).
[CrossRef]

S. Bará, J. Arines, J. Ares, and P. Prado, “Direct transformation of Zernike eye aberration coefficients between scaled, rotated and/or displaced pupils,” J. Opt. Soc. Am. A 23, 2061–2066 (2006).
[CrossRef]

Salmon, T. O.

T. O. Salmon and C. van de Pol, “Normal-eye Zernike coefficients and root-mean-square wavefront errors,” J. Cataract Refractive Surg. 32, 2064–2074 (2006).
[CrossRef]

Schwiegerling, J.

Schwiegerling, J. T.

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, and R. Webb, VSIA Standards Taskforce Members, “Standards for reporting the optical aberrations of eyes,” in Vision Science and Its Applications, V. Lakshminarayanan, ed., Vol. 35 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2000), pp. 232–244.

Shu, H.

Soloviev, O.

Tatarskii, V. I.

V. I. Tatarskii, The Propagation of Waves in the Turbulent Atmosphere (Nauka, 1967), pp. 385–390 (in Russian).

Tatulli, E.

Thibos, L. N.

L. N. Thibos, A. Bradley, and N. Lopez-Gil, “Modelling the impact of spherical aberration on accommodation,” Ophthalmic Physiolog. Opt. 33, 482–496 (2013).
[CrossRef]

L. N. Thibos, X. Hong, A. Bradley, and X. Cheng, “Statistical variation of aberration structure and image quality in a normal population of healthy eyes,” J. Opt. Soc. Am. A 19, 2329–2348 (2002).
[CrossRef]

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, and R. Webb, VSIA Standards Taskforce Members, “Standards for reporting the optical aberrations of eyes,” in Vision Science and Its Applications, V. Lakshminarayanan, ed., Vol. 35 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2000), pp. 232–244.

Thompson, K. P.

Unsbo, P.

van de Pol, C.

T. O. Salmon and C. van de Pol, “Normal-eye Zernike coefficients and root-mean-square wavefront errors,” J. Cataract Refractive Surg. 32, 2064–2074 (2006).
[CrossRef]

van Haver, S.

A. J. Janssen, S. van Haver, P. Dirksen, and J. J. Braat, “Zernike representation and Strehl ratio of optical systems with variable numerical aperture,” J. Mod. Opt. 55, 1127–1157 (2008).
[CrossRef]

Vdovin, G.

Walker, G.

Wallner, E. P.

Wang, J.

T. R. Candy and J. Wang, “Higher order monochromatic aberrations of the human infant eye,” J. Vis. 5(12):6, 543–545 (2005).
[CrossRef]

Webb, R.

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, and R. Webb, VSIA Standards Taskforce Members, “Standards for reporting the optical aberrations of eyes,” in Vision Science and Its Applications, V. Lakshminarayanan, ed., Vol. 35 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2000), pp. 232–244.

Webb, R. H.

Williams, D. R.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1998), pp. 464–466, 767–772.

Appl. Opt. (2)

Investig. Ophthalmol. Visual Sci. (1)

E. Moreno-Barriuso, J. Merayo-Lloves, S. Marcos, R. Navarro, L. Llorente, and S. Barbero, “Ocular aberrations before and after myopic corneal refractive surgery: LASIK-induced changes measured with laser ray tracing,” Investig. Ophthalmol. Visual Sci. 42, 1396–1403 (2001).

J. Cataract Refractive Surg. (1)

T. O. Salmon and C. van de Pol, “Normal-eye Zernike coefficients and root-mean-square wavefront errors,” J. Cataract Refractive Surg. 32, 2064–2074 (2006).
[CrossRef]

J. Microlithogr. Microfabr. Microsyst. (1)

A. J. Janssen and P. Dirksen, “Concise formula for the Zernike coefficients of scaled pupils,” J. Microlithogr. Microfabr. Microsyst. 5, 030501 (2006).
[CrossRef]

J. Mod. Opt. (2)

A. J. Janssen, S. van Haver, P. Dirksen, and J. J. Braat, “Zernike representation and Strehl ratio of optical systems with variable numerical aperture,” J. Mod. Opt. 55, 1127–1157 (2008).
[CrossRef]

J. A. Díaz, J. Fernández-Dorado, C. Pizarro, and J. Arasa, “Zernike coefficients for concentric, circular scaled pupils: an equivalent expression,” J. Mod. Opt. 56, 131–137 (2009).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (16)

S. Bará, “Measuring eye aberrations with Hartmann–Shack wave-front sensors: should the irradiance distribution across the eye pupil be taken into account?” J. Opt. Soc. Am. A 20, 2237–2245 (2003).
[CrossRef]

J. C. He, S. Marcos, R. H. Webb, and S. A. Burns, “Measurement of the wavefront aberration of the eye by a fast psychophysical procedure,” J. Opt. Soc. Am. A 15, 2449–2456 (1998).
[CrossRef]

K. Dillon, “Bilinear wavefront transformation,” J. Opt. Soc. Am. A 26, 1839–1846 (2009).
[CrossRef]

J. Porter, A. Guirao, I. G. Cox, and D. R. Williams, “Monochromatic aberrations of the human eye in a large population,” J. Opt. Soc. Am. A 18, 1793–1803 (2001).
[CrossRef]

L. N. Thibos, X. Hong, A. Bradley, and X. Cheng, “Statistical variation of aberration structure and image quality in a normal population of healthy eyes,” J. Opt. Soc. Am. A 19, 2329–2348 (2002).
[CrossRef]

E. Pailos, A. Ommani, L. Díaz-Santana, and S. Bará, “Centroid displacement statistics of the eye aberration,” J. Opt. Soc. Am. A 27, 1818–1827 (2010).
[CrossRef]

J. Liang, W. Grimm, S. Goelz, and J. F. Bille, “Objective measurement of the wave aberrations of the human eye using a Hartmann–Shack wave-front sensor,” J. Opt. Soc. Am. A 11, 1949–1957 (1994).
[CrossRef]

J. Liang and D. R. Williams, “Aberrations and retinal image quality of the normal human eye,” J. Opt. Soc. Am. A 14, 2873–2883 (1997).
[CrossRef]

L. Lundström and P. Unsbo, “Transformation of Zernike coefficients: scaled, translated, and rotated wavefronts with circular and elliptical pupils,” J. Opt. Soc. Am. A 24, 569–577 (2007).
[CrossRef]

E. Tatulli, “Transformation of Zernike coefficients: a Fourier-based method for scaled, translated, and rotated wavefront apertures,” J. Opt. Soc. Am. A 30, 726–732 (2013).
[CrossRef]

K. A. Goldberg and K. Geary, “Wave-front measurement errors from restricted concentric subdomains,” J. Opt. Soc. Am. A 18, 2146–2152 (2001).
[CrossRef]

J. Schwiegerling, “Scaling Zernike expansion coefficients to different pupil sizes,” J. Opt. Soc. Am. A 19, 1937–1945 (2002).
[CrossRef]

C. E. Campbell, “Matrix method to find a new set of Zernike coefficients from an original set when the aperture radius is changed,” J. Opt. Soc. Am. A 20, 209–217 (2003).
[CrossRef]

G. M. Dai, “Scaling Zernike expansion coefficients to smaller pupil sizes: a simpler formula,” J. Opt. Soc. Am. A 23, 539–543 (2006).
[CrossRef]

H. Shu, L. Luo, G. Han, and J. L. Coatrieux, “General method to derive the relationship between two sets of Zernike coefficients corresponding to different aperture sizes,” J. Opt. Soc. Am. A 23, 1960–1968 (2006).
[CrossRef]

S. Bará, J. Arines, J. Ares, and P. Prado, “Direct transformation of Zernike eye aberration coefficients between scaled, rotated and/or displaced pupils,” J. Opt. Soc. Am. A 23, 2061–2066 (2006).
[CrossRef]

J. Optom. (1)

R. Montés-Micó, P. Hernández, V. Fernández-Sánchez, S. Bonaque, F. Lara, and N. López-Gil, “Changes of the eye optics after iris constriction,” J. Optom. 3, 212–218 (2010).

J. Refract. Sur. (1)

G. M. Dai, “Validity of scaling Zernike coefficients to a larger diameter for refractive surgery,” J. Refract. Sur. 27, 837–841 (2011).
[CrossRef]

J. Vis. (1)

T. R. Candy and J. Wang, “Higher order monochromatic aberrations of the human infant eye,” J. Vis. 5(12):6, 543–545 (2005).
[CrossRef]

Ophthalmic Physiolog. Opt. (2)

L. N. Thibos, A. Bradley, and N. Lopez-Gil, “Modelling the impact of spherical aberration on accommodation,” Ophthalmic Physiolog. Opt. 33, 482–496 (2013).
[CrossRef]

J. Arines, E. Pailos, P. Prado, and S. Bará, “The contribution of the fixational eye movements to the variability of the measured ocular aberration,” Ophthalmic Physiolog. Opt. 29, 281–287 (2009).
[CrossRef]

Opt. Commun. (1)

J. Arines, P. Prado, S. Bará, and E. Acosta, “Equivalence of least-squares estimation of eye aberrations in linearly transformed reference frames,” Opt. Commun. 281, 2716–2721 (2008).
[CrossRef]

Opt. Express (3)

Opt. Lett. (3)

Optom. Vis. Sci. (1)

R. Navarro and M. A. Losada, “Aberrations and relative efficiency of light pencils in the living human eye,” Optom. Vis. Sci. 74, 540–547 (1997).
[CrossRef]

Vis. Res. (1)

J. F. Castejon-Mochon, N. Lopez-Gil, A. Benito, and P. Artal, “Ocular wave-front aberration statistics in a normal young population,” Vis. Res. 42, 1611–1617 (2002).
[CrossRef]

Other (3)

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1998), pp. 464–466, 767–772.

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, and R. Webb, VSIA Standards Taskforce Members, “Standards for reporting the optical aberrations of eyes,” in Vision Science and Its Applications, V. Lakshminarayanan, ed., Vol. 35 of OSA Trends in Optics and Photonics Series (Optical Society of America, 2000), pp. 232–244.

V. I. Tatarskii, The Propagation of Waves in the Turbulent Atmosphere (Nauka, 1967), pp. 385–390 (in Russian).

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Figures (5)

Fig. 1.
Fig. 1.

Set of eye pupils (circles) superimposed onto a Hartmann–Shack array of 19×19 microlenses. The original pupil Π of radius R is plotted with a thick red line, whereas the resized ones, of radii R=γR (γ=1.2500, 1.1875, 1.1250, 1.0625, 0.9375, 0.8750, 0.8125, and 0.7500) are drawn with thin blue lines. Red dots are placed at the centers of the 145 unvignetted microlenses used to estimate the aberrations within Π.

Fig. 2.
Fig. 2.

Mean squared overall estimation error σ2 in μm2 (black solid line with circles) versus the maximum Zernike radial order n included in the reconstruction matrix. Its three components, drawn with dashed lines, are also shown: bias (triangles), truncation (squares), and noise propagation (asterisks). The results correspond to the original pupil of radius R sampled by 145 unvignetted microlenses with a SNR=100 for a population with the aberration statistics described in the text. The overall error is minimum when the aberrations are estimated up to the radial order n=7.

Fig. 3.
Fig. 3.

Optimum value of the radial Zernike order n up to which reconstruct the aberration in resized pupils by reffiting (red lines with squares) and rescaling (blue lines with circles). Each pair of curves intersect at the original pupil (R/R=1), and they correspond, from top to bottom, to SNR=, 100, and 1, respectively.

Fig. 4.
Fig. 4.

Minimum attainable rms estimation error (in percent relative to the rms aberrations of the population, given by sqrt[trace(Ca)]), when the aberrations in the resized pupils are estimated by reffiting (red lines with squares) and rescaling (blue lines with circles). Each pair of curves intersect at the original pupil (R/R=1) and correspond, from top to bottom, to SNR=1 (upper pair), 10, 100, and (lower pair). The curves for SNR=1000, located between the last two, are not drawn for clarity.

Fig. 5.
Fig. 5.

Dependence of the minimum error and its components on changes in the population statistics and on decenterings of the eye with respect to the aberrometer. Left column: minimum attainable rms estimation errors σ for SNR=100 in case of reffiting (red lines with squares) and rescaling (blue lines with circles). Right column: relative contribution of the bias (triangles), truncation (squares), and noise propagation (asterisks) to this minimum (expressed in percent over the global quadratic error σ2). Red-dashed lines correspond to refitting, blue solid lines correspond to rescaling. First row (a), (b): population with uncorrelated Zernike modes (Ca diagonal) measured with a perfectly centered aberrometer. Second row (c), (d): population with a fully correlated Ca matrix (same diagonal terms as in the previous case and all nondiagonal Pearson coefficients equal to +1), measured with a centered aberrometer. Third row (e), (f): the first population but now measured with an aberrometer whose center is laterally shifted with respect to the eye pupil by (0.5, 0.25) units of microlens size.

Equations (6)

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σ2=i=1M(aia^i)2+i=M+1Mai2,
Cε=εεT=(aa^)(aa^)T,
σ2=trace[(IRA)Ca(IRA)T]+trace[Ca;i>M]+σν2trace[RRT],
σrefit2=trace[(IRA)Ca(IRA)T]+trace[Ca;i>M]+σν2trace[RRT],
σrefit2=trace[(IRA)TCaTT(IRA)T]+trace[TCaTT;i>M]+(R/R)2σν2trace[RRT].
σresc2=trace[(TM,MTMRA)Ca(TM,MTMRA)T]+trace[TCaTT;i>M]+σν2trace[TMRRTTMT].

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