Abstract

The time-evolution of ocular aberrations has been the subject of many studies, but so far there has been little discussion involving the modelling of the underlying temporal statistics. This paper presents a non-stationary modelling approach based on a coloured-noise generator, which can be applied to ocular aberration dynamics. The model parameters are computed from measured ocular aberration data. A custom-built aberrometer based on a Shack-Hartmann sensor was used for measurement. We present simulations based on our modelling approach, and validate them through comparison to real data. This work could be useful in areas such as the testing of ophthalmic devices and the development of improved algorithms for laser refractive surgery.

© 2010 Optical Society of America

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2010 (2)

M. Muma, D. R. Iskander, and M. J. Collins, "The role of cardiopulmonary signals in the dynamics of the eye’s wavefront aberrations," IEEE Trans. Biomed. Eng. 57, 373-383 (2010).
[CrossRef]

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] [PubMed]

2009 (1)

A. Mira-Agudelo, L. Lundström, and P. Artal, "Temporal dynamics of ocular aberrations:Monocular vs binocular vision," Ophthal. Physiol. Opt. 29, 256-263 (2009).
[CrossRef]

2008 (1)

S. O. Galetskiy, T. Yu. Cherezova, and A. V. Kudryashov, "Adaptive optics in ophthalmology: human eye wavefront generator," Proc. SPIE 6849, 68490918 (2008).

2007 (2)

2006 (2)

2005 (2)

K. Hampson, I. Munro, C. Paterson, and J. C. Dainty, "Weak correlation between the aberration dynamics of the human eye and the cardiopulmonary system," J. Opt. Soc. Am. A. 22, 1241-1250 (2005).
[CrossRef]

S. Gruppetta, F. Lacombe, and P. Puget, "Study of the dynamic aberrations of the human tear film," Opt. Express 13, 7631-7636 (2005).
[CrossRef] [PubMed]

2004 (2)

M. Zhu, M. J. Collins, and D. R. Iskander, "Microfluctuations of wavefront aberrations of the eye," Opthal. Physiol. Opt. 24, 562-571 (2004).
[CrossRef]

D. R. Iskander, M. Collins, M. Morelande, and M. Zhu, "Analyzing the dynamic wavefront aberrations in the human eye," IEEE Trans. Biomed. Eng. 51, 1969-1980 (2004).
[CrossRef] [PubMed]

2003 (2)

2002 (3)

L. N. Thibos, A. Bradley, and X. Hong, "A statistical model of the aberration structure of normal, well-corrected eyes," Ophthal. Physiol. Opt. 22, 427-433 (2002).
[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]

P. Celka and P. Colditz, "Nonlinear nonstationary Wiener model of infant seizures," IEEE Trans. Biomed. Eng. 49, 556-564 (2002).
[CrossRef] [PubMed]

2001 (2)

B. Boashash and M. Mesbah, "A time-frequency approach for newborn seizure detection," IEEE Eng. Med. Biol. Mag. 20, 54-64 (2001).
[CrossRef] [PubMed]

H. Hofer, P. Artal, and D. R. Williams, "Dynamics of the eyes wave aberration," J. Opt. Soc. Am. A. 18, 497-506 (2001).
[CrossRef]

2000 (2)

R. Navarro, E. Moreno-Barriuso, S. Bará, and T. Mancebo, "Phase plates for wave-aberration compensation in the human eye," Opt. Lett. 25, 236-238 (2000).
[CrossRef]

C. Roberts, "Future challenges to aberration-free ablative procedures," J. Refract. Surg. 16, 623-629 (2000).

1997 (1)

L. R. Stark and D. A. Atchison, "Pupil size, mean accommodation response and the fluctuations of accommodation," Opthal. Physiol. Opt. 17, 316-323 (1997).
[CrossRef]

1995 (1)

N. Kasdin, "Discrete simulation of colored noise and stochastic processes and 1/f power law noise," Proc. IEEE 83, 802-827 (1995).
[CrossRef]

1990 (1)

K. Billah and M. Shinozuka, "Numerical method for colored noise generation and its application to a bistable system," Phys. Rev. A 42, 7492-7495 (1990).
[CrossRef] [PubMed]

1982 (1)

J. D. Scargle, "Studies in astronomical time series analysis ii. statistical aspects of spectral analysis of unevenly spaced data," Astrophys. J. 263, 835-853 (1982).
[CrossRef]

1975 (2)

B. M. Hill "A simple general approach to inference about the tail of a distribution," Ann. Statist. 3, 1163-1174 (1975).
[CrossRef]

N. R. Lomb, "Least-squares frequency analysis of unequally spaced data," Astrophys. Space Sci. 39, 447-462 (1975).
[CrossRef]

Arden, G.

Artal, P.

A. Mira-Agudelo, L. Lundström, and P. Artal, "Temporal dynamics of ocular aberrations:Monocular vs binocular vision," Ophthal. Physiol. Opt. 29, 256-263 (2009).
[CrossRef]

H. Hofer, P. Artal, and D. R. Williams, "Dynamics of the eyes wave aberration," J. Opt. Soc. Am. A. 18, 497-506 (2001).
[CrossRef]

Atchison, D. A.

L. R. Stark and D. A. Atchison, "Pupil size, mean accommodation response and the fluctuations of accommodation," Opthal. Physiol. Opt. 17, 316-323 (1997).
[CrossRef]

Bará, S.

Billah, K.

K. Billah and M. Shinozuka, "Numerical method for colored noise generation and its application to a bistable system," Phys. Rev. A 42, 7492-7495 (1990).
[CrossRef] [PubMed]

Boashash, B.

L. Rankine, N. Stevenson,M. Mesbah, and B. Boashash, "A nonstationary model of newborn EEG," IEEE Trans. Biomed. Eng 54, 19-28 (2007).
[CrossRef] [PubMed]

B. Boashash and M. Mesbah, "A time-frequency approach for newborn seizure detection," IEEE Eng. Med. Biol. Mag. 20, 54-64 (2001).
[CrossRef] [PubMed]

Bradley, A.

L. N. Thibos, A. Bradley, and X. Hong, "A statistical model of the aberration structure of normal, well-corrected eyes," Ophthal. Physiol. Opt. 22, 427-433 (2002).
[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]

Celka, P.

P. Celka and P. Colditz, "Nonlinear nonstationary Wiener model of infant seizures," IEEE Trans. Biomed. Eng. 49, 556-564 (2002).
[CrossRef] [PubMed]

Cheng, X.

Cherezova, T. Yu.

S. O. Galetskiy, T. Yu. Cherezova, and A. V. Kudryashov, "Adaptive optics in ophthalmology: human eye wavefront generator," Proc. SPIE 6849, 68490918 (2008).

Colditz, P.

P. Celka and P. Colditz, "Nonlinear nonstationary Wiener model of infant seizures," IEEE Trans. Biomed. Eng. 49, 556-564 (2002).
[CrossRef] [PubMed]

Collins, M.

D. R. Iskander, M. Collins, M. Morelande, and M. Zhu, "Analyzing the dynamic wavefront aberrations in the human eye," IEEE Trans. Biomed. Eng. 51, 1969-1980 (2004).
[CrossRef] [PubMed]

Collins, M. J.

M. Muma, D. R. Iskander, and M. J. Collins, "The role of cardiopulmonary signals in the dynamics of the eye’s wavefront aberrations," IEEE Trans. Biomed. Eng. 57, 373-383 (2010).
[CrossRef]

M. Zhu, M. J. Collins, and D. R. Iskander, "Microfluctuations of wavefront aberrations of the eye," Opthal. Physiol. Opt. 24, 562-571 (2004).
[CrossRef]

Dainty, C.

Dainty, J. C.

K. Hampson, E. Mallen, and J. C. Dainty, "Coherence function analysis of the higher-order aberrations of the human eye," Opt. Lett. 31, 184-186 (2006).
[CrossRef] [PubMed]

K. Hampson, I. Munro, C. Paterson, and J. C. Dainty, "Weak correlation between the aberration dynamics of the human eye and the cardiopulmonary system," J. Opt. Soc. Am. A. 22, 1241-1250 (2005).
[CrossRef]

Davies, N.

N. Davies, L. Diaz-Santana, and D. Lara-Sucedo, "Repeatability of ocular wavefront measurement," Optom. Vis. Sci. 80, 142-150 (2003).
[CrossRef] [PubMed]

Diaz-Santana, L.

Galetskiy, S. O.

S. O. Galetskiy, T. Yu. Cherezova, and A. V. Kudryashov, "Adaptive optics in ophthalmology: human eye wavefront generator," Proc. SPIE 6849, 68490918 (2008).

Gasson, P.

Gruppetta, S.

Guériaux, V.

Hampson, K.

K. Hampson, E. Mallen, and J. C. Dainty, "Coherence function analysis of the higher-order aberrations of the human eye," Opt. Lett. 31, 184-186 (2006).
[CrossRef] [PubMed]

K. Hampson, I. Munro, C. Paterson, and J. C. Dainty, "Weak correlation between the aberration dynamics of the human eye and the cardiopulmonary system," J. Opt. Soc. Am. A. 22, 1241-1250 (2005).
[CrossRef]

Hill, B. M.

B. M. Hill "A simple general approach to inference about the tail of a distribution," Ann. Statist. 3, 1163-1174 (1975).
[CrossRef]

Hofer, H.

H. Hofer, P. Artal, and D. R. Williams, "Dynamics of the eyes wave aberration," J. Opt. Soc. Am. A. 18, 497-506 (2001).
[CrossRef]

Hong, X.

L. N. Thibos, A. Bradley, and X. Hong, "A statistical model of the aberration structure of normal, well-corrected eyes," Ophthal. Physiol. Opt. 22, 427-433 (2002).
[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]

Iskander, D. R.

M. Muma, D. R. Iskander, and M. J. Collins, "The role of cardiopulmonary signals in the dynamics of the eye’s wavefront aberrations," IEEE Trans. Biomed. Eng. 57, 373-383 (2010).
[CrossRef]

M. Zhu, M. J. Collins, and D. R. Iskander, "Microfluctuations of wavefront aberrations of the eye," Opthal. Physiol. Opt. 24, 562-571 (2004).
[CrossRef]

D. R. Iskander, M. Collins, M. Morelande, and M. Zhu, "Analyzing the dynamic wavefront aberrations in the human eye," IEEE Trans. Biomed. Eng. 51, 1969-1980 (2004).
[CrossRef] [PubMed]

Kasdin, N.

N. Kasdin, "Discrete simulation of colored noise and stochastic processes and 1/f power law noise," Proc. IEEE 83, 802-827 (1995).
[CrossRef]

Kudryashov, A. V.

S. O. Galetskiy, T. Yu. Cherezova, and A. V. Kudryashov, "Adaptive optics in ophthalmology: human eye wavefront generator," Proc. SPIE 6849, 68490918 (2008).

Lacombe, F.

Lara-Sucedo, D.

N. Davies, L. Diaz-Santana, and D. Lara-Sucedo, "Repeatability of ocular wavefront measurement," Optom. Vis. Sci. 80, 142-150 (2003).
[CrossRef] [PubMed]

Leahy, C.

Leroux, C.

Li, K. Y.

Lomb, N. R.

N. R. Lomb, "Least-squares frequency analysis of unequally spaced data," Astrophys. Space Sci. 39, 447-462 (1975).
[CrossRef]

Lundström, L.

A. Mira-Agudelo, L. Lundström, and P. Artal, "Temporal dynamics of ocular aberrations:Monocular vs binocular vision," Ophthal. Physiol. Opt. 29, 256-263 (2009).
[CrossRef]

Mallen, E.

Mancebo, T.

Mesbah, M.

L. Rankine, N. Stevenson,M. Mesbah, and B. Boashash, "A nonstationary model of newborn EEG," IEEE Trans. Biomed. Eng 54, 19-28 (2007).
[CrossRef] [PubMed]

B. Boashash and M. Mesbah, "A time-frequency approach for newborn seizure detection," IEEE Eng. Med. Biol. Mag. 20, 54-64 (2001).
[CrossRef] [PubMed]

Mira-Agudelo, A.

A. Mira-Agudelo, L. Lundström, and P. Artal, "Temporal dynamics of ocular aberrations:Monocular vs binocular vision," Ophthal. Physiol. Opt. 29, 256-263 (2009).
[CrossRef]

Morelande, M.

D. R. Iskander, M. Collins, M. Morelande, and M. Zhu, "Analyzing the dynamic wavefront aberrations in the human eye," IEEE Trans. Biomed. Eng. 51, 1969-1980 (2004).
[CrossRef] [PubMed]

Moreno-Barriuso, E.

Muma, M.

M. Muma, D. R. Iskander, and M. J. Collins, "The role of cardiopulmonary signals in the dynamics of the eye’s wavefront aberrations," IEEE Trans. Biomed. Eng. 57, 373-383 (2010).
[CrossRef]

Munro, I.

K. Hampson, I. Munro, C. Paterson, and J. C. Dainty, "Weak correlation between the aberration dynamics of the human eye and the cardiopulmonary system," J. Opt. Soc. Am. A. 22, 1241-1250 (2005).
[CrossRef]

L. Diaz-Santana, C. Torti, I. Munro, P. Gasson, and C. Dainty, "Benefit of higher closed-loop bandwidths in ocular adaptive optics," Opt. Express 11, 2597-2605 (2003).
[CrossRef] [PubMed]

Navarro, R.

Paterson, C.

K. Hampson, I. Munro, C. Paterson, and J. C. Dainty, "Weak correlation between the aberration dynamics of the human eye and the cardiopulmonary system," J. Opt. Soc. Am. A. 22, 1241-1250 (2005).
[CrossRef]

Puget, P.

Rankine, L.

L. Rankine, N. Stevenson,M. Mesbah, and B. Boashash, "A nonstationary model of newborn EEG," IEEE Trans. Biomed. Eng 54, 19-28 (2007).
[CrossRef] [PubMed]

Roberts, C.

C. Roberts, "Future challenges to aberration-free ablative procedures," J. Refract. Surg. 16, 623-629 (2000).

Scargle, J. D.

J. D. Scargle, "Studies in astronomical time series analysis ii. statistical aspects of spectral analysis of unevenly spaced data," Astrophys. J. 263, 835-853 (1982).
[CrossRef]

Shinozuka, M.

K. Billah and M. Shinozuka, "Numerical method for colored noise generation and its application to a bistable system," Phys. Rev. A 42, 7492-7495 (1990).
[CrossRef] [PubMed]

Stark, L. R.

L. R. Stark and D. A. Atchison, "Pupil size, mean accommodation response and the fluctuations of accommodation," Opthal. Physiol. Opt. 17, 316-323 (1997).
[CrossRef]

Stevenson, N.

L. Rankine, N. Stevenson,M. Mesbah, and B. Boashash, "A nonstationary model of newborn EEG," IEEE Trans. Biomed. Eng 54, 19-28 (2007).
[CrossRef] [PubMed]

Thibos, L. N.

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, A. Bradley, and X. Hong, "A statistical model of the aberration structure of normal, well-corrected eyes," Ophthal. Physiol. Opt. 22, 427-433 (2002).
[CrossRef]

Torti, C.

Williams, D. R.

H. Hofer, P. Artal, and D. R. Williams, "Dynamics of the eyes wave aberration," J. Opt. Soc. Am. A. 18, 497-506 (2001).
[CrossRef]

Yoon, G.

Zhu, M.

D. R. Iskander, M. Collins, M. Morelande, and M. Zhu, "Analyzing the dynamic wavefront aberrations in the human eye," IEEE Trans. Biomed. Eng. 51, 1969-1980 (2004).
[CrossRef] [PubMed]

M. Zhu, M. J. Collins, and D. R. Iskander, "Microfluctuations of wavefront aberrations of the eye," Opthal. Physiol. Opt. 24, 562-571 (2004).
[CrossRef]

Ann. Statist. (1)

B. M. Hill "A simple general approach to inference about the tail of a distribution," Ann. Statist. 3, 1163-1174 (1975).
[CrossRef]

Astrophys. J. (1)

J. D. Scargle, "Studies in astronomical time series analysis ii. statistical aspects of spectral analysis of unevenly spaced data," Astrophys. J. 263, 835-853 (1982).
[CrossRef]

Astrophys. Space Sci. (1)

N. R. Lomb, "Least-squares frequency analysis of unequally spaced data," Astrophys. Space Sci. 39, 447-462 (1975).
[CrossRef]

IEEE Eng. Med. Biol. Mag. (1)

B. Boashash and M. Mesbah, "A time-frequency approach for newborn seizure detection," IEEE Eng. Med. Biol. Mag. 20, 54-64 (2001).
[CrossRef] [PubMed]

IEEE Trans. Biomed. Eng (1)

L. Rankine, N. Stevenson,M. Mesbah, and B. Boashash, "A nonstationary model of newborn EEG," IEEE Trans. Biomed. Eng 54, 19-28 (2007).
[CrossRef] [PubMed]

IEEE Trans. Biomed. Eng. (3)

D. R. Iskander, M. Collins, M. Morelande, and M. Zhu, "Analyzing the dynamic wavefront aberrations in the human eye," IEEE Trans. Biomed. Eng. 51, 1969-1980 (2004).
[CrossRef] [PubMed]

M. Muma, D. R. Iskander, and M. J. Collins, "The role of cardiopulmonary signals in the dynamics of the eye’s wavefront aberrations," IEEE Trans. Biomed. Eng. 57, 373-383 (2010).
[CrossRef]

P. Celka and P. Colditz, "Nonlinear nonstationary Wiener model of infant seizures," IEEE Trans. Biomed. Eng. 49, 556-564 (2002).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. A. (2)

H. Hofer, P. Artal, and D. R. Williams, "Dynamics of the eyes wave aberration," J. Opt. Soc. Am. A. 18, 497-506 (2001).
[CrossRef]

K. Hampson, I. Munro, C. Paterson, and J. C. Dainty, "Weak correlation between the aberration dynamics of the human eye and the cardiopulmonary system," J. Opt. Soc. Am. A. 22, 1241-1250 (2005).
[CrossRef]

J. Refract. Surg. (1)

C. Roberts, "Future challenges to aberration-free ablative procedures," J. Refract. Surg. 16, 623-629 (2000).

Ophthal. Physiol. Opt. (2)

A. Mira-Agudelo, L. Lundström, and P. Artal, "Temporal dynamics of ocular aberrations:Monocular vs binocular vision," Ophthal. Physiol. Opt. 29, 256-263 (2009).
[CrossRef]

L. N. Thibos, A. Bradley, and X. Hong, "A statistical model of the aberration structure of normal, well-corrected eyes," Ophthal. Physiol. Opt. 22, 427-433 (2002).
[CrossRef]

Opt. Express (5)

Opt. Lett. (2)

Opthal. Physiol. Opt. (2)

M. Zhu, M. J. Collins, and D. R. Iskander, "Microfluctuations of wavefront aberrations of the eye," Opthal. Physiol. Opt. 24, 562-571 (2004).
[CrossRef]

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[CrossRef]

Optom. Vis. Sci. (1)

N. Davies, L. Diaz-Santana, and D. Lara-Sucedo, "Repeatability of ocular wavefront measurement," Optom. Vis. Sci. 80, 142-150 (2003).
[CrossRef] [PubMed]

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[CrossRef] [PubMed]

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[CrossRef]

Proc. SPIE (1)

S. O. Galetskiy, T. Yu. Cherezova, and A. V. Kudryashov, "Adaptive optics in ophthalmology: human eye wavefront generator," Proc. SPIE 6849, 68490918 (2008).

Other (5)

D. R. Iskander, M. R. Morelande, and M. J. Collins, "Estimating the dynamics of aberration components in the human eye," In IEEE Signal Process. Workshop Statist. pages 241-244 (2001).

E. N. Bruce, Biomedical Signal Processing and SignalModeling (Wiley Series in Telecommunications and Signal Processing, 2001).

A. Clauset, C. R. Shalizi, and M. E. J. Newman, "Power-law distributions in empirical data," arXiv:0706.1062v1, URL http://arxiv.org/abs/0706.1062v1 (2007).

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Figures (7)

Fig. 1
Fig. 1

Dynamics of selected Zernike aberrations (left) and the associated PSD estimates (right).

Fig. 2
Fig. 2

Time-frequency representation of Zernike astigmatism for subject CML (via the ZAM distribution).

Fig. 3
Fig. 3

Comparison of measured and simulated Zernike spherical aberration signals in the time-domain.

Fig. 4
Fig. 4

Comparison of measured and simulated Zernike spherical aberration signals via their estimated power spectral densities.

Fig. 5
Fig. 5

Discrete Fourier phase spectrum computed from measured data and used as the “surrogate” phase to produce the simulated aberration signal of Fig. 3. This Fourier phase spectrum representation was computed using an interpolation algorithm to replace missing data points.

Fig. 6
Fig. 6

Time-frequency coherence between real and simulated signals for Zernike spherical aberration.

Fig. 7
Fig. 7

Illustration of the relationship between statistical stability about a mean and the PSD slope at low frequencies. All signals were generated using the two-slope model, with an identical phase spectrum generated from a uniform distribution.

Equations (5)

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W ( ρ , θ ; t ) = n = 1 N m = 0 n c n m ( t ) Z n m ( ρ , θ ) + ɛ ( ρ , θ ; t )
P ( f ) c | f | γ
P ( f ) = 1 T | X ( f ) | 2
X ( f ) = | X ( f ) | e i φ ( f ) )
γ ^ ( f ) = { γ 1 if f f b r γ 2 if f > f b r

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