Abstract

Ocular monochromatic aberrations display dynamic behavior even when the eye is fixating on a stationary stimulus. The fluctuations are commonly characterized in the frequency domain using the power spectrum obtained via the Fourier transform. In this paper we used a wavelet-based multifractal analytical approach to provide a more in depth analysis of the nature of the aberration fluctuations. The aberrations of five subjects were measured at 21 Hz using an open-view Shack-Hartmann sensor. We show that the aberration dynamics are multifractal. The most frequently occurring Hölder exponent for the rms wavefront error, averaged across the five subjects, was 0.31 ± 0.10. This suggests that the time course of the aberration fluctuations is antipersistant. Future applications of multifractal analysis are discussed.

© 2011 OSA

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2011 (1)

M. Özger, “Investigating the properties of significant wave height time series using a wavelet based approach,” J. Waterw. Port Coast. Ocean Eng. 137(1), 34–42 (2011).
[CrossRef]

2010 (5)

S. Mallat and W. L. Hwang, “Singularity detection and processing using wavelets,” IEEE Trans. Inf. Theory 18, 2668–2681 (2010).

C. Leahy and C. Dainty, “A non-stationary model for simulating the dynamics of ocular aberrations,” Opt. Express 18(20), 21386–21396 (2010).
[CrossRef] [PubMed]

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(3), 2668–2681 (2010).
[CrossRef] [PubMed]

D. C. Lin and A. Sharif, “Common multifractality in the heart rate variability and brain activity of healthy humans,” Chaos 20(2), 023121 (2010).
[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(2), 373–383 (2010).
[CrossRef] [PubMed]

2009 (2)

K. M. Hampson, S. S. Chin, and E. A. H. Mallen, “Dual wavefront sensing channel monocular adaptive optics system for accommodation studies,” Opt. Express 17(20), 18229–18240 (2009).
[CrossRef] [PubMed]

A. Mira-Agudelo, L. Lundström, and P. Artal, “Temporal dynamics of ocular aberrations: monocular vs binocular vision,” Ophthalmic Physiol. Opt. 29(3), 256–263 (2009).
[CrossRef] [PubMed]

2008 (3)

S. S. Chin, K. M. Hampson, and E. A. H. Mallen, “Binocular correlation of ocular aberration dynamics,” Opt. Express 16(19), 14731–14745 (2008).
[CrossRef] [PubMed]

K. M. Hampson, S. S. Chin, and E. A. H. Mallen, “Binocular Shack-Hartmann sensor for the human eye,” J. Mod. Opt. 55(4), 703–716 (2008).
[CrossRef]

K. M. Hampson, “Adaptive optics and vision,” J. Mod. Opt. 55(21), 3425–3467 (2008).
[CrossRef]

2007 (3)

L. Diaz-Santana, V. Guériaux, G. Arden, and S. Gruppetta, “New methodology to measure the dynamics of ocular wave front aberrations during small amplitude changes of accommodation,” Opt. Express 15(9), 5649–5663 (2007).
[CrossRef] [PubMed]

J. A. Piñuela, D. Andina, K. J. McInnes, and A. M. Tarquis, “Wavelet analysis in a structured clay soil using 2-D images,” Nonlin. Processes 18, 21386–21396 (2007).

R. T. J. McAteer, C. A. Young, J. Ireland, and P. T. Gallagher, “The bursty nature of solar flare x-ray emission,” Astron. J. 662, 691–700 (2007).

2006 (3)

K. M. Hampson, E. A. H. Mallen, and C. Dainty, “Coherence function analysis of the higher-order aberrations of the human eye,” Opt. Lett. 31(2), 184–186 (2006).
[CrossRef] [PubMed]

P. Shang, Y. Lu, and S. Kama, “The application of Hölder exponent to traffic congestion warning,” Physica A 370(2), 769–776 (2006).
[CrossRef]

B. Enescu, K. Ito, and Z. R. Struzik, “Wavelet-based multiscale resolution analysis of real and simulated time-series earthquakes,” Geophys. J. Int. 164(1), 63–74 (2006).
[CrossRef]

2005 (1)

2004 (3)

L. N. Thibos, X. Hong, A. Bradley, and R. A. Applegate, “Accuracy and precision of objective refraction from wavefront aberrations,” J. Vis. 4(4), 9 (2004).
[CrossRef] [PubMed]

D. R. Iskander, M. J. Collins, M. R. Morelande, and M. Zhu, “Analyzing the dynamic wavefront aberrations in the human eye,” IEEE Trans. Biomed. Eng. 51(11), 1969–1980 (2004).
[CrossRef] [PubMed]

M. Zhu, M. J. Collins, and D. Robert Iskander, “Microfluctuations of wavefront aberrations of the eye,” Ophthalmic Physiol. Opt. 24(6), 562–571 (2004).
[CrossRef] [PubMed]

2003 (3)

2002 (3)

A. Eke, P. Herman, L. Kocsis, and L. R. Kozak, “Fractal characterization of complexity in temporal physiological signals,” Physiol. Meas. 23(1), R1 (2002).
[CrossRef] [PubMed]

A. L. Goldberger, L. A. N. Amaral, J. M. Hausdorff, P. Ch. Ivanov, C.-K. Peng, and H. E. Stanley, “Fractal dynamics in physiology: alterations with disease and aging,” Proc. Natl. Acad. Sci. U.S.A. 99(90001Suppl 1), 2466–2472 (2002).
[CrossRef] [PubMed]

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, and R. Webb, “Standards for reporting the optical aberrations of eyes,” Refract. Surg. 18, 652–660 (2002).

2001 (2)

P. C. Ivanov, L. A. Nunes Amaral, A. L. Goldberger, S. Havlin, M. G. Rosenblum, H. E. Stanley, and Z. R. Struzik, “From 1/f noise to multifractal cascades in heartbeat dynamics,” Chaos 11(3), 641–652 (2001).
[CrossRef] [PubMed]

H. Hofer, P. Artal, B. Singer, J. L. Aragón, and D. R. Williams, “Dynamics of the eye’s wave aberration,” J. Opt. Soc. Am. A 18(3), 497–506 (2001).
[CrossRef] [PubMed]

1999 (1)

P. C. Ivanov, L. A. Amaral, A. L. Goldberger, S. Havlin, M. G. Rosenblum, Z. R. Struzik, and H. E. Stanley, “Multifractality in human heartbeat dynamics,” Nature 399(6735), 461–465 (1999).
[CrossRef] [PubMed]

1998 (1)

C. Torrence and G. P. Compo, “A practical guide to wavelet analysis,” Bull. Am. Met. Soc. 79(1), 61–78 (1998).
[CrossRef]

1995 (1)

A. Arneodo, E. Bacry, and J. Muzy, “The thermodynamics of fractals revisited with wavelets,” Physica A 213(1-2), 232–275 (1995).
[CrossRef]

1992 (1)

B. Telfer and H. H. Szu, “New wavelet transform normalization to remove frequency bias,” Opt. Eng. 31(9), 1830–1834 (1992).
[CrossRef]

Amaral, L. A.

P. C. Ivanov, L. A. Amaral, A. L. Goldberger, S. Havlin, M. G. Rosenblum, Z. R. Struzik, and H. E. Stanley, “Multifractality in human heartbeat dynamics,” Nature 399(6735), 461–465 (1999).
[CrossRef] [PubMed]

Amaral, L. A. N.

A. L. Goldberger, L. A. N. Amaral, J. M. Hausdorff, P. Ch. Ivanov, C.-K. Peng, and H. E. Stanley, “Fractal dynamics in physiology: alterations with disease and aging,” Proc. Natl. Acad. Sci. U.S.A. 99(90001Suppl 1), 2466–2472 (2002).
[CrossRef] [PubMed]

Andina, D.

J. A. Piñuela, D. Andina, K. J. McInnes, and A. M. Tarquis, “Wavelet analysis in a structured clay soil using 2-D images,” Nonlin. Processes 18, 21386–21396 (2007).

Applegate, R. A.

L. N. Thibos, X. Hong, A. Bradley, and R. A. Applegate, “Accuracy and precision of objective refraction from wavefront aberrations,” J. Vis. 4(4), 9 (2004).
[CrossRef] [PubMed]

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, and R. Webb, “Standards for reporting the optical aberrations of eyes,” Refract. Surg. 18, 652–660 (2002).

Aragón, J. L.

Arden, G.

Arneodo, A.

A. Arneodo, E. Bacry, and J. Muzy, “The thermodynamics of fractals revisited with wavelets,” Physica A 213(1-2), 232–275 (1995).
[CrossRef]

Artal, P.

A. Mira-Agudelo, L. Lundström, and P. Artal, “Temporal dynamics of ocular aberrations: monocular vs binocular vision,” Ophthalmic Physiol. Opt. 29(3), 256–263 (2009).
[CrossRef] [PubMed]

H. Hofer, P. Artal, B. Singer, J. L. Aragón, and D. R. Williams, “Dynamics of the eye’s wave aberration,” J. Opt. Soc. Am. A 18(3), 497–506 (2001).
[CrossRef] [PubMed]

Bacry, E.

A. Arneodo, E. Bacry, and J. Muzy, “The thermodynamics of fractals revisited with wavelets,” Physica A 213(1-2), 232–275 (1995).
[CrossRef]

Bille, J.

Bradley, A.

L. N. Thibos, X. Hong, A. Bradley, and R. A. Applegate, “Accuracy and precision of objective refraction from wavefront aberrations,” J. Vis. 4(4), 9 (2004).
[CrossRef] [PubMed]

Chin, S. S.

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(2), 373–383 (2010).
[CrossRef] [PubMed]

D. R. Iskander, M. J. Collins, M. R. Morelande, and M. Zhu, “Analyzing the dynamic wavefront aberrations in the human eye,” IEEE Trans. Biomed. Eng. 51(11), 1969–1980 (2004).
[CrossRef] [PubMed]

M. Zhu, M. J. Collins, and D. Robert Iskander, “Microfluctuations of wavefront aberrations of the eye,” Ophthalmic Physiol. Opt. 24(6), 562–571 (2004).
[CrossRef] [PubMed]

Compo, G. P.

C. Torrence and G. P. Compo, “A practical guide to wavelet analysis,” Bull. Am. Met. Soc. 79(1), 61–78 (1998).
[CrossRef]

Dainty, C.

Diaz-Santana, L.

Eke, A.

A. Eke, P. Herman, L. Kocsis, and L. R. Kozak, “Fractal characterization of complexity in temporal physiological signals,” Physiol. Meas. 23(1), R1 (2002).
[CrossRef] [PubMed]

Enescu, B.

B. Enescu, K. Ito, and Z. R. Struzik, “Wavelet-based multiscale resolution analysis of real and simulated time-series earthquakes,” Geophys. J. Int. 164(1), 63–74 (2006).
[CrossRef]

Gallagher, P. T.

R. T. J. McAteer, C. A. Young, J. Ireland, and P. T. Gallagher, “The bursty nature of solar flare x-ray emission,” Astron. J. 662, 691–700 (2007).

Gasson, P.

Goldberger, A. L.

A. L. Goldberger, L. A. N. Amaral, J. M. Hausdorff, P. Ch. Ivanov, C.-K. Peng, and H. E. Stanley, “Fractal dynamics in physiology: alterations with disease and aging,” Proc. Natl. Acad. Sci. U.S.A. 99(90001Suppl 1), 2466–2472 (2002).
[CrossRef] [PubMed]

P. C. Ivanov, L. A. Nunes Amaral, A. L. Goldberger, S. Havlin, M. G. Rosenblum, H. E. Stanley, and Z. R. Struzik, “From 1/f noise to multifractal cascades in heartbeat dynamics,” Chaos 11(3), 641–652 (2001).
[CrossRef] [PubMed]

P. C. Ivanov, L. A. Amaral, A. L. Goldberger, S. Havlin, M. G. Rosenblum, Z. R. Struzik, and H. E. Stanley, “Multifractality in human heartbeat dynamics,” Nature 399(6735), 461–465 (1999).
[CrossRef] [PubMed]

Gray, L. S.

D. Seidel, L. S. Gray, and G. Heron, “Retinotopic accommodation responses in myopia,” Invest. Ophthalmol. Vis. Sci. 44(3), 1035–1041 (2003).
[CrossRef] [PubMed]

Gruppetta, S.

Guériaux, V.

Hampson, K. M.

Hausdorff, J. M.

A. L. Goldberger, L. A. N. Amaral, J. M. Hausdorff, P. Ch. Ivanov, C.-K. Peng, and H. E. Stanley, “Fractal dynamics in physiology: alterations with disease and aging,” Proc. Natl. Acad. Sci. U.S.A. 99(90001Suppl 1), 2466–2472 (2002).
[CrossRef] [PubMed]

Havlin, S.

P. C. Ivanov, L. A. Nunes Amaral, A. L. Goldberger, S. Havlin, M. G. Rosenblum, H. E. Stanley, and Z. R. Struzik, “From 1/f noise to multifractal cascades in heartbeat dynamics,” Chaos 11(3), 641–652 (2001).
[CrossRef] [PubMed]

P. C. Ivanov, L. A. Amaral, A. L. Goldberger, S. Havlin, M. G. Rosenblum, Z. R. Struzik, and H. E. Stanley, “Multifractality in human heartbeat dynamics,” Nature 399(6735), 461–465 (1999).
[CrossRef] [PubMed]

Herman, P.

A. Eke, P. Herman, L. Kocsis, and L. R. Kozak, “Fractal characterization of complexity in temporal physiological signals,” Physiol. Meas. 23(1), R1 (2002).
[CrossRef] [PubMed]

Heron, G.

D. Seidel, L. S. Gray, and G. Heron, “Retinotopic accommodation responses in myopia,” Invest. Ophthalmol. Vis. Sci. 44(3), 1035–1041 (2003).
[CrossRef] [PubMed]

Hofer, H.

Hong, X.

L. N. Thibos, X. Hong, A. Bradley, and R. A. Applegate, “Accuracy and precision of objective refraction from wavefront aberrations,” J. Vis. 4(4), 9 (2004).
[CrossRef] [PubMed]

Hwang, W. L.

S. Mallat and W. L. Hwang, “Singularity detection and processing using wavelets,” IEEE Trans. Inf. Theory 18, 2668–2681 (2010).

Ireland, J.

R. T. J. McAteer, C. A. Young, J. Ireland, and P. T. Gallagher, “The bursty nature of solar flare x-ray emission,” Astron. J. 662, 691–700 (2007).

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(2), 373–383 (2010).
[CrossRef] [PubMed]

D. R. Iskander, M. J. Collins, M. R. Morelande, and M. Zhu, “Analyzing the dynamic wavefront aberrations in the human eye,” IEEE Trans. Biomed. Eng. 51(11), 1969–1980 (2004).
[CrossRef] [PubMed]

Ito, K.

B. Enescu, K. Ito, and Z. R. Struzik, “Wavelet-based multiscale resolution analysis of real and simulated time-series earthquakes,” Geophys. J. Int. 164(1), 63–74 (2006).
[CrossRef]

Ivanov, P. C.

P. C. Ivanov, L. A. Nunes Amaral, A. L. Goldberger, S. Havlin, M. G. Rosenblum, H. E. Stanley, and Z. R. Struzik, “From 1/f noise to multifractal cascades in heartbeat dynamics,” Chaos 11(3), 641–652 (2001).
[CrossRef] [PubMed]

P. C. Ivanov, L. A. Amaral, A. L. Goldberger, S. Havlin, M. G. Rosenblum, Z. R. Struzik, and H. E. Stanley, “Multifractality in human heartbeat dynamics,” Nature 399(6735), 461–465 (1999).
[CrossRef] [PubMed]

Ivanov, P. Ch.

A. L. Goldberger, L. A. N. Amaral, J. M. Hausdorff, P. Ch. Ivanov, C.-K. Peng, and H. E. Stanley, “Fractal dynamics in physiology: alterations with disease and aging,” Proc. Natl. Acad. Sci. U.S.A. 99(90001Suppl 1), 2466–2472 (2002).
[CrossRef] [PubMed]

Kama, S.

P. Shang, Y. Lu, and S. Kama, “The application of Hölder exponent to traffic congestion warning,” Physica A 370(2), 769–776 (2006).
[CrossRef]

Kocsis, L.

A. Eke, P. Herman, L. Kocsis, and L. R. Kozak, “Fractal characterization of complexity in temporal physiological signals,” Physiol. Meas. 23(1), R1 (2002).
[CrossRef] [PubMed]

Kozak, L. R.

A. Eke, P. Herman, L. Kocsis, and L. R. Kozak, “Fractal characterization of complexity in temporal physiological signals,” Physiol. Meas. 23(1), R1 (2002).
[CrossRef] [PubMed]

Leahy, C.

Leroux, C.

Lin, D. C.

D. C. Lin and A. Sharif, “Common multifractality in the heart rate variability and brain activity of healthy humans,” Chaos 20(2), 023121 (2010).
[CrossRef] [PubMed]

Lu, Y.

P. Shang, Y. Lu, and S. Kama, “The application of Hölder exponent to traffic congestion warning,” Physica A 370(2), 769–776 (2006).
[CrossRef]

Lundström, L.

A. Mira-Agudelo, L. Lundström, and P. Artal, “Temporal dynamics of ocular aberrations: monocular vs binocular vision,” Ophthalmic Physiol. Opt. 29(3), 256–263 (2009).
[CrossRef] [PubMed]

Mallat, S.

S. Mallat and W. L. Hwang, “Singularity detection and processing using wavelets,” IEEE Trans. Inf. Theory 18, 2668–2681 (2010).

Mallen, E. A. H.

McAteer, R. T. J.

R. T. J. McAteer, C. A. Young, J. Ireland, and P. T. Gallagher, “The bursty nature of solar flare x-ray emission,” Astron. J. 662, 691–700 (2007).

McInnes, K. J.

J. A. Piñuela, D. Andina, K. J. McInnes, and A. M. Tarquis, “Wavelet analysis in a structured clay soil using 2-D images,” Nonlin. Processes 18, 21386–21396 (2007).

Mira-Agudelo, A.

A. Mira-Agudelo, L. Lundström, and P. Artal, “Temporal dynamics of ocular aberrations: monocular vs binocular vision,” Ophthalmic Physiol. Opt. 29(3), 256–263 (2009).
[CrossRef] [PubMed]

Morelande, M. R.

D. R. Iskander, M. J. Collins, M. R. Morelande, and M. Zhu, “Analyzing the dynamic wavefront aberrations in the human eye,” IEEE Trans. Biomed. Eng. 51(11), 1969–1980 (2004).
[CrossRef] [PubMed]

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(2), 373–383 (2010).
[CrossRef] [PubMed]

Munro, I.

Muzy, J.

A. Arneodo, E. Bacry, and J. Muzy, “The thermodynamics of fractals revisited with wavelets,” Physica A 213(1-2), 232–275 (1995).
[CrossRef]

Nirmaier, T.

Nunes Amaral, L. A.

P. C. Ivanov, L. A. Nunes Amaral, A. L. Goldberger, S. Havlin, M. G. Rosenblum, H. E. Stanley, and Z. R. Struzik, “From 1/f noise to multifractal cascades in heartbeat dynamics,” Chaos 11(3), 641–652 (2001).
[CrossRef] [PubMed]

Özger, M.

M. Özger, “Investigating the properties of significant wave height time series using a wavelet based approach,” J. Waterw. Port Coast. Ocean Eng. 137(1), 34–42 (2011).
[CrossRef]

Paterson, C.

Peng, C.-K.

A. L. Goldberger, L. A. N. Amaral, J. M. Hausdorff, P. Ch. Ivanov, C.-K. Peng, and H. E. Stanley, “Fractal dynamics in physiology: alterations with disease and aging,” Proc. Natl. Acad. Sci. U.S.A. 99(90001Suppl 1), 2466–2472 (2002).
[CrossRef] [PubMed]

Piñuela, J. A.

J. A. Piñuela, D. Andina, K. J. McInnes, and A. M. Tarquis, “Wavelet analysis in a structured clay soil using 2-D images,” Nonlin. Processes 18, 21386–21396 (2007).

Pudasaini, G.

Robert Iskander, D.

M. Zhu, M. J. Collins, and D. Robert Iskander, “Microfluctuations of wavefront aberrations of the eye,” Ophthalmic Physiol. Opt. 24(6), 562–571 (2004).
[CrossRef] [PubMed]

Rosenblum, M. G.

P. C. Ivanov, L. A. Nunes Amaral, A. L. Goldberger, S. Havlin, M. G. Rosenblum, H. E. Stanley, and Z. R. Struzik, “From 1/f noise to multifractal cascades in heartbeat dynamics,” Chaos 11(3), 641–652 (2001).
[CrossRef] [PubMed]

P. C. Ivanov, L. A. Amaral, A. L. Goldberger, S. Havlin, M. G. Rosenblum, Z. R. Struzik, and H. E. Stanley, “Multifractality in human heartbeat dynamics,” Nature 399(6735), 461–465 (1999).
[CrossRef] [PubMed]

Schwiegerling, J. T.

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, and R. Webb, “Standards for reporting the optical aberrations of eyes,” Refract. Surg. 18, 652–660 (2002).

Seidel, D.

D. Seidel, L. S. Gray, and G. Heron, “Retinotopic accommodation responses in myopia,” Invest. Ophthalmol. Vis. Sci. 44(3), 1035–1041 (2003).
[CrossRef] [PubMed]

Shang, P.

P. Shang, Y. Lu, and S. Kama, “The application of Hölder exponent to traffic congestion warning,” Physica A 370(2), 769–776 (2006).
[CrossRef]

Sharif, A.

D. C. Lin and A. Sharif, “Common multifractality in the heart rate variability and brain activity of healthy humans,” Chaos 20(2), 023121 (2010).
[CrossRef] [PubMed]

Singer, B.

Stanley, H. E.

A. L. Goldberger, L. A. N. Amaral, J. M. Hausdorff, P. Ch. Ivanov, C.-K. Peng, and H. E. Stanley, “Fractal dynamics in physiology: alterations with disease and aging,” Proc. Natl. Acad. Sci. U.S.A. 99(90001Suppl 1), 2466–2472 (2002).
[CrossRef] [PubMed]

P. C. Ivanov, L. A. Nunes Amaral, A. L. Goldberger, S. Havlin, M. G. Rosenblum, H. E. Stanley, and Z. R. Struzik, “From 1/f noise to multifractal cascades in heartbeat dynamics,” Chaos 11(3), 641–652 (2001).
[CrossRef] [PubMed]

P. C. Ivanov, L. A. Amaral, A. L. Goldberger, S. Havlin, M. G. Rosenblum, Z. R. Struzik, and H. E. Stanley, “Multifractality in human heartbeat dynamics,” Nature 399(6735), 461–465 (1999).
[CrossRef] [PubMed]

Struzik, Z. R.

B. Enescu, K. Ito, and Z. R. Struzik, “Wavelet-based multiscale resolution analysis of real and simulated time-series earthquakes,” Geophys. J. Int. 164(1), 63–74 (2006).
[CrossRef]

P. C. Ivanov, L. A. Nunes Amaral, A. L. Goldberger, S. Havlin, M. G. Rosenblum, H. E. Stanley, and Z. R. Struzik, “From 1/f noise to multifractal cascades in heartbeat dynamics,” Chaos 11(3), 641–652 (2001).
[CrossRef] [PubMed]

P. C. Ivanov, L. A. Amaral, A. L. Goldberger, S. Havlin, M. G. Rosenblum, Z. R. Struzik, and H. E. Stanley, “Multifractality in human heartbeat dynamics,” Nature 399(6735), 461–465 (1999).
[CrossRef] [PubMed]

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B. Telfer and H. H. Szu, “New wavelet transform normalization to remove frequency bias,” Opt. Eng. 31(9), 1830–1834 (1992).
[CrossRef]

Tarquis, A. M.

J. A. Piñuela, D. Andina, K. J. McInnes, and A. M. Tarquis, “Wavelet analysis in a structured clay soil using 2-D images,” Nonlin. Processes 18, 21386–21396 (2007).

Telfer, B.

B. Telfer and H. H. Szu, “New wavelet transform normalization to remove frequency bias,” Opt. Eng. 31(9), 1830–1834 (1992).
[CrossRef]

Thibos, L. N.

L. N. Thibos, X. Hong, A. Bradley, and R. A. Applegate, “Accuracy and precision of objective refraction from wavefront aberrations,” J. Vis. 4(4), 9 (2004).
[CrossRef] [PubMed]

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, and R. Webb, “Standards for reporting the optical aberrations of eyes,” Refract. Surg. 18, 652–660 (2002).

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C. Torrence and G. P. Compo, “A practical guide to wavelet analysis,” Bull. Am. Met. Soc. 79(1), 61–78 (1998).
[CrossRef]

Torti, C.

Webb, R.

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, and R. Webb, “Standards for reporting the optical aberrations of eyes,” Refract. Surg. 18, 652–660 (2002).

Williams, D. R.

Young, C. A.

R. T. J. McAteer, C. A. Young, J. Ireland, and P. T. Gallagher, “The bursty nature of solar flare x-ray emission,” Astron. J. 662, 691–700 (2007).

Zhu, M.

D. R. Iskander, M. J. Collins, M. R. Morelande, and M. Zhu, “Analyzing the dynamic wavefront aberrations in the human eye,” IEEE Trans. Biomed. Eng. 51(11), 1969–1980 (2004).
[CrossRef] [PubMed]

M. Zhu, M. J. Collins, and D. Robert Iskander, “Microfluctuations of wavefront aberrations of the eye,” Ophthalmic Physiol. Opt. 24(6), 562–571 (2004).
[CrossRef] [PubMed]

Astron. J. (1)

R. T. J. McAteer, C. A. Young, J. Ireland, and P. T. Gallagher, “The bursty nature of solar flare x-ray emission,” Astron. J. 662, 691–700 (2007).

Bull. Am. Met. Soc. (1)

C. Torrence and G. P. Compo, “A practical guide to wavelet analysis,” Bull. Am. Met. Soc. 79(1), 61–78 (1998).
[CrossRef]

Chaos (2)

P. C. Ivanov, L. A. Nunes Amaral, A. L. Goldberger, S. Havlin, M. G. Rosenblum, H. E. Stanley, and Z. R. Struzik, “From 1/f noise to multifractal cascades in heartbeat dynamics,” Chaos 11(3), 641–652 (2001).
[CrossRef] [PubMed]

D. C. Lin and A. Sharif, “Common multifractality in the heart rate variability and brain activity of healthy humans,” Chaos 20(2), 023121 (2010).
[CrossRef] [PubMed]

Geophys. J. Int. (1)

B. Enescu, K. Ito, and Z. R. Struzik, “Wavelet-based multiscale resolution analysis of real and simulated time-series earthquakes,” Geophys. J. Int. 164(1), 63–74 (2006).
[CrossRef]

IEEE Trans. Biomed. Eng. (2)

D. R. Iskander, M. J. Collins, M. R. Morelande, and M. Zhu, “Analyzing the dynamic wavefront aberrations in the human eye,” IEEE Trans. Biomed. Eng. 51(11), 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(2), 373–383 (2010).
[CrossRef] [PubMed]

IEEE Trans. Inf. Theory (1)

S. Mallat and W. L. Hwang, “Singularity detection and processing using wavelets,” IEEE Trans. Inf. Theory 18, 2668–2681 (2010).

Invest. Ophthalmol. Vis. Sci. (1)

D. Seidel, L. S. Gray, and G. Heron, “Retinotopic accommodation responses in myopia,” Invest. Ophthalmol. Vis. Sci. 44(3), 1035–1041 (2003).
[CrossRef] [PubMed]

J. Mod. Opt. (2)

K. M. Hampson, “Adaptive optics and vision,” J. Mod. Opt. 55(21), 3425–3467 (2008).
[CrossRef]

K. M. Hampson, S. S. Chin, and E. A. H. Mallen, “Binocular Shack-Hartmann sensor for the human eye,” J. Mod. Opt. 55(4), 703–716 (2008).
[CrossRef]

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

J. Vis. (1)

L. N. Thibos, X. Hong, A. Bradley, and R. A. Applegate, “Accuracy and precision of objective refraction from wavefront aberrations,” J. Vis. 4(4), 9 (2004).
[CrossRef] [PubMed]

J. Waterw. Port Coast. Ocean Eng. (1)

M. Özger, “Investigating the properties of significant wave height time series using a wavelet based approach,” J. Waterw. Port Coast. Ocean Eng. 137(1), 34–42 (2011).
[CrossRef]

Nature (1)

P. C. Ivanov, L. A. Amaral, A. L. Goldberger, S. Havlin, M. G. Rosenblum, Z. R. Struzik, and H. E. Stanley, “Multifractality in human heartbeat dynamics,” Nature 399(6735), 461–465 (1999).
[CrossRef] [PubMed]

Nonlin. Processes (1)

J. A. Piñuela, D. Andina, K. J. McInnes, and A. M. Tarquis, “Wavelet analysis in a structured clay soil using 2-D images,” Nonlin. Processes 18, 21386–21396 (2007).

Ophthalmic Physiol. Opt. (2)

M. Zhu, M. J. Collins, and D. Robert Iskander, “Microfluctuations of wavefront aberrations of the eye,” Ophthalmic Physiol. Opt. 24(6), 562–571 (2004).
[CrossRef] [PubMed]

A. Mira-Agudelo, L. Lundström, and P. Artal, “Temporal dynamics of ocular aberrations: monocular vs binocular vision,” Ophthalmic Physiol. Opt. 29(3), 256–263 (2009).
[CrossRef] [PubMed]

Opt. Eng. (1)

B. Telfer and H. H. Szu, “New wavelet transform normalization to remove frequency bias,” Opt. Eng. 31(9), 1830–1834 (1992).
[CrossRef]

Opt. Express (7)

Opt. Lett. (1)

Physica A (2)

A. Arneodo, E. Bacry, and J. Muzy, “The thermodynamics of fractals revisited with wavelets,” Physica A 213(1-2), 232–275 (1995).
[CrossRef]

P. Shang, Y. Lu, and S. Kama, “The application of Hölder exponent to traffic congestion warning,” Physica A 370(2), 769–776 (2006).
[CrossRef]

Physiol. Meas. (1)

A. Eke, P. Herman, L. Kocsis, and L. R. Kozak, “Fractal characterization of complexity in temporal physiological signals,” Physiol. Meas. 23(1), R1 (2002).
[CrossRef] [PubMed]

Proc. Natl. Acad. Sci. U.S.A. (1)

A. L. Goldberger, L. A. N. Amaral, J. M. Hausdorff, P. Ch. Ivanov, C.-K. Peng, and H. E. Stanley, “Fractal dynamics in physiology: alterations with disease and aging,” Proc. Natl. Acad. Sci. U.S.A. 99(90001Suppl 1), 2466–2472 (2002).
[CrossRef] [PubMed]

Refract. Surg. (1)

L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, and R. Webb, “Standards for reporting the optical aberrations of eyes,” Refract. Surg. 18, 652–660 (2002).

Other (4)

C. D. Cutler, “A review of the theory and estimation of fractal dimension,” in Dimension Estimation and Models, H. Tong, ed. (World Scientific Publishing Co Pte Ltd, 1993).

P. S. Addison, The Illustrated Wavelet Transform Handbook (Taylor and Francis, 2002).

J. C. Van den Berg, Wavelets in Physics (Cambridge University Press, 2004).

S. Mallat, A Wavelet Tour of Signal Processing: the Sparse Way (Academic Press, 2009).

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Figures (11)

Fig. 1
Fig. 1

The meaning of fractal dimension Dbox as calculated using the box counting dimension. It can be argued that a square is not a true fractal as it does not have a fractional dimension. It is included here merely to give an intuitive illustration of fractal dimension.

Fig. 2
Fig. 2

(a) An example of the time course of the rms wavefront error of one subject measured over a 6 mm pupil and sampled at 21 Hz. (b) A section of the signal in (a) scaled using Eqs. (4) and (5). Hurst exponent = 0.25.

Fig. 3
Fig. 3

Shack-Hartmann sensor used for measurement of aberrations. PM: plane mirror; BS: beamsplitter; L: lens. Focal length is in millimeters.

Fig. 4
Fig. 4

A typical singularity spectrum for a multifractal time series. For a monofractal time series the singularity spectrum would be a single point.

Fig. 5
Fig. 5

How the CWT is formed. For each wavelet position τ, and wavelet width (scale) s, the wavelet is convolved with the signal to form a point in the CWT. The Mexican hat wavelet is shown.

Fig. 6
Fig. 6

Analytical procedure to obtain the singularity spectrum. (a) The time evolution of the rms wavefront error for JC. (b) The continuous wavelet transform. (c) The modulus-maxima map. (d) Log of the partition function versus log of the scale. (e) The tau mass exponents obtained from the slope of the lines in (d). (f) The singularity spectrum obtained from (e) and Eq. (10).

Fig. 7
Fig. 7

CWT for the rms wavefront error each subject. The plots have been normalized for each subject.

Fig. 8
Fig. 8

CWT of each aberration for YP.

Fig. 9
Fig. 9

rms wavefront error singularity spectrum for each subject.

Fig. 10
Fig. 10

Average most frequently found Hölder exponent across subjects for each aberration. Error bars represent ± 1 S.D.

Fig. 11
Fig. 11

Power spectrum of the rms wavefront error for each subject.

Tables (1)

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Table 1 Subject demographics

Equations (14)

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

P S D ( f ) 1 f β ,
D b o x = lim ε 0 log N ε log ( 1 / ε ) ,
D H u r s t = 2 H f B m ,
x λ x ,
y λ H f B m y .
H f B m = β 1 2 .
Z ( ε , q ) = i = 1 N ε μ i ( ε ) q ,
τ ( q ) = lim s 0 log Z ( ε , q ) log ε ,
Z ( ε , q ) ε τ ( q ) .
D H a u s ( h ) = q . h ( q ) τ ( q ) ,
h ( q ) = d τ ( q ) d q .
C W T x ( τ , s ) = 1 s x ( t ) Ψ * ( t τ s ) d t ,
f s c a l e = f c s Δ T ,
Z ( s , q ) = i = 1 N m | C W T ( s , m i ) | q ,

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