## Multifractal nature of ocular aberration dynamics of the human eye |

Biomedical Optics Express, Vol. 2, Issue 3, pp. 464-470 (2011)

http://dx.doi.org/10.1364/BOE.2.000464

Acrobat PDF (1103 KB)

### 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

## 1. Introduction

1. 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]

*PSD(f)*is the power spectral density at a given frequency

*f*, and

*β*is the spectral index.

*β*is determined from a linear regression of log

*PSD(f)*on log

*f*. For the human eye, the spectral index of the time evolution of rms wavefront error has been found to be in the region of 1.1-1.5 [1

1. 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]

5. 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]

6. 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]

7. 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]

9. 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]

10. 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]

*D.*The larger the fractal dimension, the more the time series fills the Euclidean space it is contained in, and the higher the degree of irregularity [11]. There are several definitions of

*D*and several methods are used to calculate it. One example is the so-called box counting dimension, wherewhere

*N*is the number of non-overlapping boxes of length

_{ε}*ε*required to cover the time series [6

6. 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]

*D*is shown in Fig. 1 for the case of a square object. The procedure for a time series is fundamentally the same.

_{box}7. 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]

9. 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]

*H*, and consequently

*D*, may vary in time. Hence a more in depth analytical approach is required to characterize such time series. As ocular aberration dynamics show some correlation with the heart beat [13

13. 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]

15. 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]

## 2. Aberration measurements

^{2}. Further details of the system can be found at [4

4. 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]

16. 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]

16. 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]

16. 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]

17. 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]

## 3. Fractal analysis

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

*h*, which is inversely proportional to the strength of the singularity. The Hölder exponent reflects the rate of decay of the amplitude of the fluctuations in the time series in the neighborhood of the time location that is being analyzed [20

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

21. 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]

21. 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]

*H*becomes

*h*in Eq. (3).

*h*. An example of such a spectrum is shown in Fig. 4 . The

*x*-axis represents the Hölder exponent (inverse of singularity strength), and the

*y*-axis represents the Hausdorff dimension

*D*.

_{Haus}*D*represents how completely each Hölder exponent fills the space (time series) it is embedded in. Hence

_{Haus}*D*(

_{Haus}*h*) is the fractal dimension of the singularities characterized by

*h*[22]. For a time series, the maximum possible value of

*D*(

_{Haus}*h*) is one. This would indicate that the singularity represented by

*h*is present everywhere in the signal [23,24

24. 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]

### 3.1 Overview of singularity spectrum determination

### 3.2 Singularity spectrum determination via wavelets

#### 3.2.1 Continuous wavelet transform

*x(t)*is given bywhich constitutes a convolution of the signal

*x(t)*with a wavelet

*Ψ*that is translated by

*τ*and scaled (stretched) by

*s*. The scale is inversely proportional to frequency. The

*1/s*normalization factor has been used as it removes any bias associated with frequency [26

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

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

#### 3.2.1.1 Practical implementation

*cwt*function of the Matlab Wavelet Toolbox, (Matlab version R2009a, Wavelet Toolbox version 4.4). Following this, the modulus of each coefficient was calculated. It was these coefficients that were used throughout the analysis. Figure 6b shows the CWT coefficients of the rms wavefront error (Fig. 6a) for subject JC. The corresponding frequency represented by each scale

*f*, is also shown. This was calculated using the Matlab function

_{scale}*scal2frq*in whichwhere f

_{c}is an estimate of the frequency of the mother (unscaled) wavelet found from the dominant peak in the Fourier based power spectrum of the wavelet, s is scale, and ΔT is the time between samples. For the Mexican hat wavelet f

_{c}is 0.25 Hz. The minimum scale (maximum frequency) used in the analysis was the closest integer scale corresponding to when f

_{s}was the Nyquist frequency (10.5 Hz). The maximum scale (minimum frequency) was the closest integer scale corresponding to when f

_{s}was the inverse of the length of the signal: 0.04 Hz for JC, EM and CS, 0.08 Hz for KH and CV.

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

#### 3.2.2 Wavelet transform modulus maximum map

*s*, point (

*s*,

*τ*) is a local maximum if |CWT(

_{max}*s*,

*τ*)|<|CWT(

*s*,

*τ*)| when

_{max}*τ*is either to the left or right of

*τ*and |CWT(

_{max}*s*,

*τ*)| ≤|CWT(

*s*,

*τ*)| the other side [28]. Following this the maxima are chained across scales to form modulus maxima lines. It can be shown that the locations of the singularities are the

_{max}*τ*values where the maxima lines converge at small scales [28,29]. Only maxima belonging to these chains are used in the calculation of the partition function. From Eq. (7), the partition function becomeswhere CWT(

*s,m*) is the coefficient of the maxima

_{i}*i*at scale

*s*, and

*N*is the number of maxima at that scale. In effect, the locations of the modulus maxima tell us where to position our ‘boxes’ to capture the singular behavior of the time series [19

_{m}19. A. Arneodo, E. Bacry, and J. Muzy, “The thermodynamics of fractals revisited with wavelets,” Physica A **213**(1-2), 232–275 (1995). [CrossRef]

#### 3.2.2.1 Practical implementation

*q*< 0, small values of the CWT result in divergences in the partition function. To prevent this, for each chain, the CWT values are tracked from low to high scale and at each scale the supremum is taken [19

**213**(1-2), 232–275 (1995). [CrossRef]

#### 3.2.3 Spectrum determination

*τ(q)*spectrum that was determined using Eq. (8). The Hölder exponents are given by the slope of the

*τ(q)*spectrum (Eq. (11)). Finally the spectrum is calculated using Eq. (10). The singularity spectrum for the rms wavefront error for JC is shown in Fig. 6f. The

*q*values used were −3 to + 3.

### 3.3 Power spectrum analysis

*β*can be used to classify a signal and determine its fractal dimension. When doing so it is recommended that the high frequency components in the range

*f*/8 <

_{s}*f*<

_{s}*f*/2, where

_{s}*f*is the sampling frequency, are excluded [6

_{s}6. 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]

## 4. Results

*h*(

*q*= 0), averaged across subjects is 0.31 ± 0.10. Using Eq. (3), this corresponds to an average fractal dimension (

*D*) of 1.69. Figure 10 shows the most frequently found Hölder exponent for each aberration averaged across subjects.

_{Hurst}## 5. Discussion

### 5.1 Comparison with previous studies

1. 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]

5. 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]

30. 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]

31. 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]

32. 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]

### 5.2 Fractal nature of aberration dynamics

30. 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]

31. 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]

7. 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]

10. 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]

### 5.3 Aberration dynamics, the heartbeat and accommodation

13. 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]

15. 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]

10. 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]

### 5.4 Potential limitations owing to measurement resolution and time length of the data

### 5.5 Future applications of multifractal analysis for ocular aberration

31. 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]

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

**99**(90001Suppl 1), 2466–2472 (2002). [CrossRef] [PubMed]

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

## 6. Conclusion

## Acknowledgments

## References and links

1. | 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 |

2. | L. Diaz-Santana, C. Torti, I. Munro, P. Gasson, and C. Dainty, “Benefit of higher closed-loop bandwidths in ocular adaptive optics,” Opt. Express |

3. | T. Nirmaier, G. Pudasaini, and J. Bille, “Very fast wave-front measurements at the human eye with a custom CMOS-based Hartmann-Shack sensor,” Opt. Express |

4. | S. S. Chin, K. M. Hampson, and E. A. H. Mallen, “Binocular correlation of ocular aberration dynamics,” Opt. Express |

5. | A. Mira-Agudelo, L. Lundström, and P. Artal, “Temporal dynamics of ocular aberrations: monocular vs binocular vision,” Ophthalmic Physiol. Opt. |

6. | A. Eke, P. Herman, L. Kocsis, and L. R. Kozak, “Fractal characterization of complexity in temporal physiological signals,” Physiol. Meas. |

7. | 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 |

8. | 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 |

9. | D. C. Lin and A. Sharif, “Common multifractality in the heart rate variability and brain activity of healthy humans,” Chaos |

10. | 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. |

11. | C. D. Cutler, “A review of the theory and estimation of fractal dimension,” in |

12. | P. S. Addison, |

13. | M. Zhu, M. J. Collins, and D. Robert Iskander, “Microfluctuations of wavefront aberrations of the eye,” Ophthalmic Physiol. Opt. |

14. | K. M. Hampson, I. Munro, C. Paterson, and C. Dainty, “Weak correlation between the aberration dynamics of the human eye and the cardiopulmonary system,” J. Opt. Soc. Am. A |

15. | 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. |

16. | K. M. Hampson, S. S. Chin, and E. A. H. Mallen, “Binocular Shack-Hartmann sensor for the human eye,” J. Mod. Opt. |

17. | K. M. Hampson, S. S. Chin, and E. A. H. Mallen, “Dual wavefront sensing channel monocular adaptive optics system for accommodation studies,” Opt. Express |

18. | L. N. Thibos, R. A. Applegate, J. T. Schwiegerling, and R. Webb, “Standards for reporting the optical aberrations of eyes,” Refract. Surg. |

19. | A. Arneodo, E. Bacry, and J. Muzy, “The thermodynamics of fractals revisited with wavelets,” Physica A |

20. | P. Shang, Y. Lu, and S. Kama, “The application of Hölder exponent to traffic congestion warning,” Physica A |

21. | B. Enescu, K. Ito, and Z. R. Struzik, “Wavelet-based multiscale resolution analysis of real and simulated time-series earthquakes,” Geophys. J. Int. |

22. | J. C. Van den Berg, |

23. | R. T. J. McAteer, C. A. Young, J. Ireland, and P. T. Gallagher, “The bursty nature of solar flare x-ray emission,” Astron. J. |

24. | M. Özger, “Investigating the properties of significant wave height time series using a wavelet based approach,” J. Waterw. Port Coast. Ocean Eng. |

25. | 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 |

26. | B. Telfer and H. H. Szu, “New wavelet transform normalization to remove frequency bias,” Opt. Eng. |

27. | C. Torrence and G. P. Compo, “A practical guide to wavelet analysis,” Bull. Am. Met. Soc. |

28. | S. Mallat and W. L. Hwang, “Singularity detection and processing using wavelets,” IEEE Trans. Inf. Theory |

29. | S. Mallat, |

30. | 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. |

31. | C. Leahy and C. Dainty, “A non-stationary model for simulating the dynamics of ocular aberrations,” Opt. Express |

32. | 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 |

33. | L. N. Thibos, X. Hong, A. Bradley, and R. A. Applegate, “Accuracy and precision of objective refraction from wavefront aberrations,” J. Vis. |

34. | K. M. Hampson, E. A. H. Mallen, and C. Dainty, “Coherence function analysis of the higher-order aberrations of the human eye,” Opt. Lett. |

35. | 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 |

36. | K. M. Hampson, “Adaptive optics and vision,” J. Mod. Opt. |

37. | D. Seidel, L. S. Gray, and G. Heron, “Retinotopic accommodation responses in myopia,” Invest. Ophthalmol. Vis. Sci. |

**OCIS Codes**

(330.4875) Vision, color, and visual optics : Optics of physiological systems

(330.7326) Vision, color, and visual optics : Visual optics, modeling

**ToC Category:**

Vision, Color, and Visual Optics

**History**

Original Manuscript: November 12, 2010

Revised Manuscript: January 18, 2011

Manuscript Accepted: January 26, 2011

Published: February 1, 2011

**Citation**

Karen M. Hampson and Edward A. H. Mallen, "Multifractal nature of ocular aberration dynamics of the human eye," Biomed. Opt. Express **2**, 464-470 (2011)

http://www.opticsinfobase.org/boe/abstract.cfm?URI=boe-2-3-464

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### References

- 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]
- 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(20), 2597–2605 (2003). [CrossRef] [PubMed]
- T. Nirmaier, G. Pudasaini, and J. Bille, “Very fast wave-front measurements at the human eye with a custom CMOS-based Hartmann-Shack sensor,” Opt. Express 11(21), 2704–2716 (2003). [CrossRef] [PubMed]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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).
- 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]
- K. M. Hampson, I. Munro, C. Paterson, and C. Dainty, “Weak correlation between the aberration dynamics of the human eye and the cardiopulmonary system,” J. Opt. Soc. Am. A 22(7), 1241–1250 (2005). [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]
- 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, 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]
- 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).
- 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]
- 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]
- J. C. Van den Berg, Wavelets in Physics (Cambridge University Press, 2004).
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