Abstract

The purpose of this work is to outline a simple model to assess the relative merits of different sampling grids for ocular aberrometry and illustrate it with an example. While in traditional Hartmann-Shack setups the sampling grid geometries have been somewhat restricted by the geometries of the available microlens arrays, other techniques such as laser ray tracing or spatially resolved refractometry allow for a greater freedom of choice. For all available setups, including HS, it is worth studying which of these choices perform better in terms of accuracy (closeness of the obtained results to the actual ones) and precision (uncertainty of the obtained results). Whilst the mathematical model presented in this paper is quite general and it can be applied to optimise existing or new aberrometers, the numerical results presented in the example are only valid for the particular aberration sample used and centroiding algorithms studied, and should not be generalised outside of these boundaries.

© 2005 Optical Society of America

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References

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  1. P. B. Liebelt, An Introduction to Optimal Estimation, (Addison-Wesley, Reading, MA, 1967) pg. 138.
  2. 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]
  3. B. P. Medoff, "Image reconstruction from limited data: theory and applications in computerized tomography," in Image Recovery: Theory and Application, H. Stark, ed. (Academic, New York, 1987), Chap. 9.
  4. J. Herrmann, �??Least-squares wave front errors of minimum norm,�?? J. Opt. Soc. Am. 70, 28�??35 (1980).
    [CrossRef]
  5. W.H. Southwell, �??Wave-front estimation from wave-front slope measurements,�?? J. Opt. Soc. Am. 70, 998�??1006 (1980).
    [CrossRef]
  6. L. Diaz Santana Haro, "Wavefront sensing in the human eye with a Shack-Hartmann sensor," Ph.D. thesis (Imperial College of Science Technology and Medicine, 2000).
  7. J. Schwiegerling, �??Scaling Zernike expansion coefficients to different pupil sizes,�?? J. Opt. Soc. Am. A 19, 1937�??1945 (2002).
    [CrossRef]

Image Recovery: Theory and Application (1)

B. P. Medoff, "Image reconstruction from limited data: theory and applications in computerized tomography," in Image Recovery: Theory and Application, H. Stark, ed. (Academic, New York, 1987), Chap. 9.

J. Opt. Soc. Am. (2)

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

Other (2)

L. Diaz Santana Haro, "Wavefront sensing in the human eye with a Shack-Hartmann sensor," Ph.D. thesis (Imperial College of Science Technology and Medicine, 2000).

P. B. Liebelt, An Introduction to Optimal Estimation, (Addison-Wesley, Reading, MA, 1967) pg. 138.

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Figures (6)

Fig. 1.
Fig. 1.

First M′ = 35 columns of the coupling matrix RA for a square grid of 69 square microlenses assuming that only the first M=20 Zernike modes are included in the estimator R. (a)with R constructed using the correct model of measurements (wavefront slopes spatially averaged over each subpupil); (b)with an incorrect measurement model (wavefront slopes evaluated at the center of each subpupil). The values are shown in a logarithmic grayscale comprising four decades (white = 1, black < 0.0001).

Fig. 2.
Fig. 2.

Different geometries analysed. From left to right: Square, Hexagonal, Polar2, Polar4 and Polar6.

Fig. 3.
Fig. 3.

noise propagators for different geometries and number of modes

Fig. 4.
Fig. 4.

(a) Centroding error as a function of spot intensity for different levels of electronic noise. (b) Comparison between α and β. Each row depicts a level of electronic noise, from 2 to 32 electrons rms from the top row to the bottom one. Each column depicts a different number of reconstructed modes and each symbol a different geometry. For each geometry the column furthest to the left corresponds with 14 modes (4th order) increasing to the right up to 44 modes (8th order). The diagonal line represents α= β

Fig. 5.
Fig. 5.

Bias introduced for the different samples as a function of grid density. (a) Square grid, (b) Hexagonal grid, (c) Polar grid.

Fig. 6.
Fig. 6.

Bias introduced for the different samples as a function of grid density. (a) Very low density, (b) Low density, (c) Medium density, (d) High density and (e) Very high density.

Tables (2)

Tables Icon

Table 1. Values of α for different geometries and different number of modes. As the number of modes increases so does the value of α. Also note that α takes maximum values for a square geometry and minimum for polar4. See text for further details

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Table 2. Values of β for several different levels of electronic noise. Note that β increases together with the electronic noise

Equations (42)

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m s = P s ( r ) s W ( r ) d 2 r + ν s
I ( r ) a I ( r ) d 2 r
W ( r ) = i = 1 M a i Z i ( r )
m = Aa + ν
P s ( r ) s Z i ( r ) d 2 r
W e ( r ) = i = 1 M a ̂ i Z i ( r )
a ̂ = Rm
a ̂ = R ( Aa + ν ) = RAa + R ν
< a ̂ > = < RAa + R ν > = RA < a >
< a a ̂ > = ( I RA ) < a >
< a ̂ > = RAa
< a a ̂ > = ( I RA ) a
A = ( A T A ) 1 A T
R = ( A T A ) 1 A T
σ 2 ( a ) = Π [ W ( r ; a ) W e ( r ; a ) ] 2 d 2 r
σ 2 = σ 2 ( a ) = Π [ W ( r ; a ) W e ( r ; a ) ] 2 d 2 r
= i = 1 M ( a ̂ i a i ) 2 + i = M + 1 M ( a i ) 2
C fg = < fg T >
C e = < ( a ̂ a ) ( a ̂ a ) T >
C e = < a ̂ a ̂ T > < a ̂ a T > < a a ̂ T > + < a a T >
= R [ A C a A T + A C + C T A T + C ν ] R T R [ A C a T + C T ]
[ C a A T + C ] R T + C a
C e = ( I RA ) C a ( I RA ) T + R C ν R T
C e = ( I RA ) C a ( I RA ) T + σ ν 2 R R T
C e = σ ν 2 R R T = σ ν 2 [ ( A T A ) 1 A T ] [ ( A T A ) 1 A T ] T
= σ ν 2 [ ( A T A ) 1 A T ] [ A ( A T A ) 1 ] = σ ν 2 ( A T A ) 1
σ ν 2 ( A M T A M ) 1
σ ν 2 Tr [ ( A M T A M ) 1 ]
Tr [ ( I RA ) C a ( I RA ) T ]
S n = Mn ; n > = 2
S n = 1 ; n = 1 ;
σ 2 = σ ν 2 Tr ( A M T A M ) 1 )
= σ ν 2 Tr ( N σ )
= σ ν 2 N σ
N σ = A x α
σ ν 2 = D I β
I = C x
N σ = A ( C I ) α
= B I α
σ 2 = σ ν 2 N σ
= Γ x β α
σ 2 = Γ = constant

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