Abstract

We discuss a method for the study of the spatial statistics of the ocular aberrations, based on the direct use of the Hartmann–Shack centroid displacements, avoiding the wavefront reconstruction step. Centroid diagrams are introduced as a helpful aid to visualize basic properties of the aberration datasets, and slope-related second-order statistical functions are applied to check the compatibility between the experimental data and different models for the aberration statistics. Preliminary results suggest that no single power-law spectrum (e.g., Kolmogorov’s) is able to represent the whole range of spatial statistics of individual eye fluctuations and that more elaborated models, including at least the contribution of a relevant defocus fluctuation term, are required. This centroid-based approach allows for an easier intercomparison of results between laboratories and avoids the bias and information loss associated with the estimation of a reduced number of Zernike coefficients from a much wider slope data set.

© 2010 Optical Society of America

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    [CrossRef]
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  6. J. F. Castejón-Mochón, N. López-Gil, A. Benito, and P. Artal, “Ocular wave-front aberration statistics in a normal young population,” Vision Res. 42, 1611–1617 (2002).
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  8. M. Zhu, M. J. Collins, and D. R. Iskander, “Microfluctuations of wavefront aberrations of the eye,” Ophthalmic Physiol. Opt. 24, 562–571 (2004).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  31. J. Liang, B. Grimm, S. Goelz, and J. F. Bille, “Objective measurement of wave aberrations of the human eye with the use of a Hartmann–Shack wave-front sensor,” J. Opt. Soc. Am. A 11, 1949–1957 (1994).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2009

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

2007

2006

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] [PubMed]

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]

2005

2004

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, 1969–1980 (2004).
[CrossRef] [PubMed]

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

2003

2002

2001

2000

1999

V. V. Voitsekhovich and S. Bará, “Efficiency of optimum Kolmogorov estimators for different atmospheric statistics: Hartmann test,” Opt. Commun. 165, 163–170 (1999).
[CrossRef]

C. H. Rao, W. H. Jiang, and N. Ling, “Measuring the power-law exponent of an atmospheric turbulence phase power spectrum with a Shack–Hartmann wave-front sensor,” Opt. Lett. 24, 1008–1010 (1999).
[CrossRef]

1998

V. V. Voitsekhovich, S. Bará, S. Ríos, and E. Acosta, “Minimum-variance phase reconstruction from Hartmann sensors with circular subpupils,” Opt. Commun. 148, 225–229 (1998).
[CrossRef]

1997

1996

1995

1994

P. A. Bakut, V. E. Kirakosyants, V. A. Loginov, C. J. Solomon, and J. C. Dainty, “Optimal wavefront reconstruction from a Shack–Hartmann sensor by use of a Bayesian algorithm,” Opt. Commun. 109, 10–15 (1994).
[CrossRef]

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

G. Cao and X. Yu, “Accuracy analysis of a Hartmann–Shack wavefront sensor operated with a faint object,” Opt. Eng. (Bellingham) 33, 2331–2335 (1994).
[CrossRef]

1992

M. C. Roggemann, “Optical-performance of fully and partially compensated adaptive optics systems using least-squares and minimum variance phase reconstructors,” Comput. Electr. Eng. 18, 451–466 (1992).
[CrossRef]

1989

1983

1981

1976

Acosta, E.

V. V. Voitsekhovich, S. Bará, S. Ríos, and E. Acosta, “Minimum-variance phase reconstruction from Hartmann sensors with circular subpupils,” Opt. Commun. 148, 225–229 (1998).
[CrossRef]

Applegate, R. A.

L. Thibos, R. A. Applegate, J. T. Schwiegerling, and R. Webb, in Vision Science and Its Applications, V.Lakshminarayanan, ed. (Optical Society of America, 2000), pp. 232–244.

Aragón, J. L.

Arden, G.

Ares, J.

Arines, J.

Artal, P.

Bakut, P. A.

P. A. Bakut, V. E. Kirakosyants, V. A. Loginov, C. J. Solomon, and J. C. Dainty, “Optimal wavefront reconstruction from a Shack–Hartmann sensor by use of a Bayesian algorithm,” Opt. Commun. 109, 10–15 (1994).
[CrossRef]

Bará, S.

Barrett, H. H.

H. H. Barrett and K. Myers, Foundations of Image Science (Wiley-Interscience, 2004).

Benito, A.

J. F. Castejón-Mochón, N. López-Gil, A. Benito, and P. Artal, “Ocular wave-front aberration statistics in a normal young population,” Vision Res. 42, 1611–1617 (2002).
[CrossRef] [PubMed]

Bille, J. F.

Boreman, G. D.

Born, M.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge U. Press, 1999).
[PubMed]

Bradley, A.

Cagigal, M. P.

Canales, V. F.

Cao, G.

G. Cao and X. Yu, “Accuracy analysis of a Hartmann–Shack wavefront sensor operated with a faint object,” Opt. Eng. (Bellingham) 33, 2331–2335 (1994).
[CrossRef]

Castejón-Mochón, J. F.

M. P. Cagigal, V. F. Canales, J. F. Castejón-Mochón, P. M. Prieto, N. López-Gil, and P. Artal, “Statistical description of wave-front aberration in the human eye,” Opt. Lett. 27, 37–39 (2002).
[CrossRef]

J. F. Castejón-Mochón, N. López-Gil, A. Benito, and P. Artal, “Ocular wave-front aberration statistics in a normal young population,” Vision Res. 42, 1611–1617 (2002).
[CrossRef] [PubMed]

Cheng, X.

Collins, M. J.

M. Zhu, M. J. Collins, and D. R. Iskander, “Microfluctuations of wavefront aberrations of the eye,” Ophthalmic Physiol. Opt. 24, 562–571 (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, 1969–1980 (2004).
[CrossRef] [PubMed]

Cox, I. G.

Dainty, C.

Dainty, J. C.

T. W. Nicholls, G. D. Boreman, and J. C. Dainty, “Use of a Shack–Hartmann wave-front sensor to measure deviations from a Kolmogorov phase spectrum,” Opt. Lett. 20, 2460–2462 (1995).
[CrossRef] [PubMed]

P. A. Bakut, V. E. Kirakosyants, V. A. Loginov, C. J. Solomon, and J. C. Dainty, “Optimal wavefront reconstruction from a Shack–Hartmann sensor by use of a Bayesian algorithm,” Opt. Commun. 109, 10–15 (1994).
[CrossRef]

Diaz-Santana, L.

Frieden, B. R.

B. R. Frieden, Probability, Statistical Optics and Data Testing: A Problem Solving Approach (Springer-Verlag, 1991).

Gardner, C. S.

Goelz, S.

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, 2000).

Grimm, B.

Gruppetta, S.

Guériaux, V.

Guirao, A.

Hampson, K. M.

Hardy, J. W.

J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford U. Press, 1998), pp. 147–150, 273.

Herrmann, J.

Hofer, H.

Hong, X.

Iroshnikov, N. G.

A. V. Larichev, P. V. Ivanov, N. G. Iroshnikov, and V. I. Shmal’gauzen, “Measurement of eye aberrations in a speckle field,” Quantum Electron. 31, 1108–1112 (2001).
[CrossRef]

Iskander, D. R.

M. Zhu, M. J. Collins, and D. R. Iskander, “Microfluctuations of wavefront aberrations of the eye,” Ophthalmic Physiol. Opt. 24, 562–571 (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, 1969–1980 (2004).
[CrossRef] [PubMed]

Ivanov, P. V.

A. V. Larichev, P. V. Ivanov, N. G. Iroshnikov, and V. I. Shmal’gauzen, “Measurement of eye aberrations in a speckle field,” Quantum Electron. 31, 1108–1112 (2001).
[CrossRef]

Jiang, W. H.

Kirakosyants, V. E.

P. A. Bakut, V. E. Kirakosyants, V. A. Loginov, C. J. Solomon, and J. C. Dainty, “Optimal wavefront reconstruction from a Shack–Hartmann sensor by use of a Bayesian algorithm,” Opt. Commun. 109, 10–15 (1994).
[CrossRef]

Larichev, A. V.

A. V. Larichev, P. V. Ivanov, N. G. Iroshnikov, and V. I. Shmal’gauzen, “Measurement of eye aberrations in a speckle field,” Quantum Electron. 31, 1108–1112 (2001).
[CrossRef]

Liang, J.

Ling, N.

Loginov, V. A.

P. A. Bakut, V. E. Kirakosyants, V. A. Loginov, C. J. Solomon, and J. C. Dainty, “Optimal wavefront reconstruction from a Shack–Hartmann sensor by use of a Bayesian algorithm,” Opt. Commun. 109, 10–15 (1994).
[CrossRef]

López-Gil, N.

M. P. Cagigal, V. F. Canales, J. F. Castejón-Mochón, P. M. Prieto, N. López-Gil, and P. Artal, “Statistical description of wave-front aberration in the human eye,” Opt. Lett. 27, 37–39 (2002).
[CrossRef]

J. F. Castejón-Mochón, N. López-Gil, A. Benito, and P. Artal, “Ocular wave-front aberration statistics in a normal young population,” Vision Res. 42, 1611–1617 (2002).
[CrossRef] [PubMed]

Lundström, L.

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

Mira-Agudelo, A.

A. Mira-Agudelo, L. Lundström, and P. Artal, “Temporal dynamics of ocular aberrations: monocular vs binocular vision,” Ophthalmic Physiol. Opt. 29, 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, 1969–1980 (2004).
[CrossRef] [PubMed]

Munro, I.

Myers, K.

H. H. Barrett and K. Myers, Foundations of Image Science (Wiley-Interscience, 2004).

Nicholls, T. W.

Noll, R. J.

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, 1991).

Paterson, C.

Porter, J.

Prado, P.

Prieto, P. M.

Rao, C. H.

Ríos, S.

V. V. Voitsekhovich, S. Bará, S. Ríos, and E. Acosta, “Minimum-variance phase reconstruction from Hartmann sensors with circular subpupils,” Opt. Commun. 148, 225–229 (1998).
[CrossRef]

Roggemann, M. C.

E. E. Silbaugh, B. M. Welsh, and M. C. Roggemann, “Characterization of atmospheric turbulence phase statistics using wave-front slope measurements,” J. Opt. Soc. Am. A 13, 2453–2460 (1996).
[CrossRef]

M. C. Roggemann, “Optical-performance of fully and partially compensated adaptive optics systems using least-squares and minimum variance phase reconstructors,” Comput. Electr. Eng. 18, 451–466 (1992).
[CrossRef]

M. C. Roggemann and B. Welsh, Imaging Through Turbulence (CRC, 1996).

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. T.

L. Thibos, R. A. Applegate, J. T. Schwiegerling, and R. Webb, in Vision Science and Its Applications, V.Lakshminarayanan, ed. (Optical Society of America, 2000), pp. 232–244.

Shmal’gauzen, V. I.

A. V. Larichev, P. V. Ivanov, N. G. Iroshnikov, and V. I. Shmal’gauzen, “Measurement of eye aberrations in a speckle field,” Quantum Electron. 31, 1108–1112 (2001).
[CrossRef]

Silbaugh, E. E.

E. E. Silbaugh, B. M. Welsh, and M. C. Roggemann, “Characterization of atmospheric turbulence phase statistics using wave-front slope measurements,” J. Opt. Soc. Am. A 13, 2453–2460 (1996).
[CrossRef]

E. E. Silbaugh, “Characterization of atmospheric turbulence over long horizontal paths using optical slope measurements,” Master’s thesis (U.S. Air Force Institute of Technology, 1995).

Singer, B.

Solomon, C. J.

P. A. Bakut, V. E. Kirakosyants, V. A. Loginov, C. J. Solomon, and J. C. Dainty, “Optimal wavefront reconstruction from a Shack–Hartmann sensor by use of a Bayesian algorithm,” Opt. Commun. 109, 10–15 (1994).
[CrossRef]

Soloviev, O.

Thibos, L.

L. Thibos, R. A. Applegate, J. T. Schwiegerling, and R. Webb, in Vision Science and Its Applications, V.Lakshminarayanan, ed. (Optical Society of America, 2000), pp. 232–244.

Thibos, L. N.

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]

Vargas-Martín, F.

Vdovin, G.

Voitsekhovich, V. V.

V. V. Voitsekhovich and S. Bará, “Efficiency of optimum Kolmogorov estimators for different atmospheric statistics: Hartmann test,” Opt. Commun. 165, 163–170 (1999).
[CrossRef]

V. V. Voitsekhovich, S. Bará, S. Ríos, and E. Acosta, “Minimum-variance phase reconstruction from Hartmann sensors with circular subpupils,” Opt. Commun. 148, 225–229 (1998).
[CrossRef]

Walker, G.

Wallner, E. P.

Webb, R.

L. Thibos, R. A. Applegate, J. T. Schwiegerling, and R. Webb, in Vision Science and Its Applications, V.Lakshminarayanan, ed. (Optical Society of America, 2000), pp. 232–244.

Welsh, B.

M. C. Roggemann and B. Welsh, Imaging Through Turbulence (CRC, 1996).

Welsh, B. M.

Williams, D. R.

Wolf, E.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge U. Press, 1999).
[PubMed]

Yu, X.

G. Cao and X. Yu, “Accuracy analysis of a Hartmann–Shack wavefront sensor operated with a faint object,” Opt. Eng. (Bellingham) 33, 2331–2335 (1994).
[CrossRef]

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, 1969–1980 (2004).
[CrossRef] [PubMed]

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

Comput. Electr. Eng.

M. C. Roggemann, “Optical-performance of fully and partially compensated adaptive optics systems using least-squares and minimum variance phase reconstructors,” Comput. Electr. Eng. 18, 451–466 (1992).
[CrossRef]

IEEE Trans. Biomed. Eng.

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, 1969–1980 (2004).
[CrossRef] [PubMed]

J. Cataract Refractive Surg.

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. Opt. Soc. Am.

J. Opt. Soc. Am. A

S. Bará, “Characteristic functions of Hartmann–Shack wavefront sensors and laser-ray-tracing aberrometers,” J. Opt. Soc. Am. A 24, 3700–3707 (2007).
[CrossRef]

B. M. Welsh and C. S. Gardner, “Performance analysis of adaptive-optics systems using laser guide stars and slope sensors,” J. Opt. Soc. Am. A 6, 1913–1923 (1989).
[CrossRef]

E. E. Silbaugh, B. M. Welsh, and M. C. Roggemann, “Characterization of atmospheric turbulence phase statistics using wave-front slope measurements,” J. Opt. Soc. Am. A 13, 2453–2460 (1996).
[CrossRef]

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]

P. M. Prieto, F. Vargas-Martín, S. Goelz, and P. Artal, “Analysis of the performance of the Hartmann–Shack sensor in the human eye,” J. Opt. Soc. Am. A 17, 1388–1398 (2000).
[CrossRef]

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

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, 1241–1250 (2005).
[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]

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]

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]

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Figures (5)

Fig. 1
Fig. 1

Two examples of centroid diagrams obtained with a 37-subpupil square microlens array, showing the centroid clouds corresponding to two eyes with different degrees of homogeneity and isotropy (centroid displacements magnified 5×).

Fig. 2
Fig. 2

Centroid diagrams for (a) eye 1 and (b) eye 2, which belong to the groups of young- and middle-age people, respectively. The fluctuations of the different centroid positions with respect to each subpupil average were magnified by a factor of 7 for clarity. Insets: enlarged view of several centroid clouds of each eye.

Fig. 3
Fig. 3

Experimental values (black marks) of the centroid variances for (a) eye 1 and (b) eye 2. In (a) the values of the total variance of the centroid displacements of eye 1 at each microlens are plotted. The parabolic curve of the defocus model was fitted to the data (solid line). The abscissa ζ s is the distance from the center of the s th microlens to the vertex of the parabola, which is located at the estimated average eye pupil position. Graph (b) represents the variance of the x-component of the centroid displacements of eye 2 against the radial coordinate of the subpupil centers. The solid line is equal to the average of the variance over microlenses. The distance between the two dashed lines is equal to twice the standard deviation of the variance values over microlenses.

Fig. 4
Fig. 4

Experimental LSSF (black circles) and TSSF (white diamonds) for the x-component of the centroid displacements for (a) eye 1 and (b) eye 2, plotted against the normalized distance l between microlenses. The error bars represent the value of the sample standard deviations. The solid lines in (a) correspond to a least-squares fit of the theoretical LSSF (upper curve) and TSSF (lower curve) for an eye model whose statistics are dominated by defocus fluctuations. The solid lines in (b) correspond to a least-squares fit of the theoretical LSSF (upper curve) and TSSF (lower curve) for a Kolmogorov’s power-law eye model.

Fig. 5
Fig. 5

Tau parameters of the analyzed eyes. (a) Eye 1: experimental values (black circles) and Lorentzian curve predicted by the defocus-dominated eye model (solid line), with parameters extracted from the fits of Fig. 4a. (b) Eye 2: experimental values (black circles) and several power-law models (solid lines) with different exponents β. The error bars represent the sample standard deviations.

Equations (17)

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μ ( r s ) = Ω ( r , r s ) W ( r ) d 2 r ,
μ ( r s ) = [ Ω ( r , r s ) ] W ( r ) d 2 r ,
Ω ( r ) = 1 d 2 rect ( r d ) = { d 2 , | x | d / 2 , | y | d / 2 0 , otherwise . }
μ u ( r s ) = [ Ω ( r r s ) / u ] W ( r ) d 2 r .
m ( r s ) = μ ( r s ) + ν ( r s ) .
D W ( r 1 , r 2 ) = [ W ( r 1 ) W ( r 2 ) ] 2 ,
μ ̃ u 2 = 1 2 d 2 r 1 d 2 r 2 Ω ( r 1 ) u 1 Ω ( r 2 ) u 2 D W ̃ ( r 1 r 2 ) ,
D μ ̃ u ( r s , r t ) = [ μ ̃ u ( r s ) μ ̃ u ( r t ) ] 2 ,
D ̂ μ ̃ u ( r s , r t ) = 1 K k = 1 K [ m ̃ u ( k ) ( r s ) m ̃ u ( k ) ( r t ) ] 2 D ν u ( r s , r t ) ,
D W ̃ ( r 1 r 2 ) = γ β ( r 1 r 2 / ρ 0 ) β 2 ,
W s ( r ) = ( 2 3 / R 2 ) a 4 r b 2
m ̃ u ( r s ) = [ a a ] u s [ a b u a b u ] + ν u ( r s ) .
m ̃ u 2 ( r s ) = σ a 2 [ u s b u ] 2 + σ b u 2 a 2 + ν u 2 ( r s ) ,
D m ̃ u ( r s , r t ) = σ a 2 ( u s u t ) 2 + 2 σ ν 2 .
L m ̃ u ( l ) = σ a 2 d 2 l 2 + 2 σ ν 2 ,
T m ̃ u ( l ) = 2 σ ν 2 ,
τ m ̃ u ( l ) = [ 1 + ( σ a 2 d 2 / 2 σ ν 2 ) l 2 ] 1 ,

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