Abstract

Adaptive optics (AO) has been recently used for the development of ophthalmic devices. Its main objective has been to obtain high-resolution images for diagnostic purposes or to estimate high-order eye aberrations. The core of every AO system is an optical device that is able to modify the wavefront shape of the light entering the system; if you know the shape of the incoming wavefront, it is possible to correct the aberrations introduced in the optical path from the source to the image. The aim of this paper is to demonstrate the feasibility, although in a simulated system, of estimating and correcting an aberrated wavefront shape by means of an iterative gradient-descent-like software procedure, acting on a point source image, without expensive wavefront sensors or the burdensome computation of the point-spread-function (PSF) of the optical system. In such a way, it is possible to obtain a speed and repeatability advantage over classical stochastic algorithms. A hierarchy in the aberrations is introduced, in order to reduce the dimensionality of the state space to be searched. The proposed algorithm is tested on a simple optical system that has been simulated with ray-tracing software, with randomly generated aberrations, and compared with a recently proposed algorithm for wavefront sensorless adaptive optics.

© 2007 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2007 (2)

M. J. Booth, "Wavefront sensorless adaptive optics for large aberrations," Opt. Lett. 32, 5-7 (2007).
[CrossRef]

G. Naletto, F. Frassetto, N. Codogno, E. Grisan, S. Bonora, V. Da Deppo, and A. Ruggeri, "No wavefront sensor adaptive optics system for compensation of primary aberrations by software analysis of a point source image. 2. Tests," Appl. Opt. 46, 0000-0000 (2007). [same issue (81590)]
[CrossRef]

2006 (3)

2005 (1)

L. Murray, J. C. Dainty, and E. Daly, "Wavefront correction through image sharpness maximisation," Proc. SPIE 5823, 40-47 (2005).
[CrossRef]

2004 (3)

F. Fankhauser, P. F. Niederer, S. Kwasniewska, and F. van der Zypen, "Supernormal vision, high-resolution retinal imaging, multiphoton imaging and nanosurgery of the cornea-a review," Technol. Health Care 12, 443-453 (2004).

M. Glanc, E. Gendron, F. Lacombe, D. Lafaille, J. F. Le Gargasson, and P. Léna, "Towards wide-field retinal imaging with adaptive optics," Opt. Commun. 230, 225-238 (2004).
[CrossRef]

N. Doble and D. R. Williams, "The application of MEMS technology for adaptive optics in vision science," IEEE J. Sel. Top. Quantum Electron. 10, 629-635 (2004).
[CrossRef]

2003 (2)

W. Quan, Z.-Q. Wang, G.-G. Mu, and L. Ning, "Correction of the aberrations in the human eyes with svag1 thin-film transistor liquid-crystal display," Optik 114, 467-471 (2003).
[CrossRef]

J. R. Fienup and J. J. Miller, "Aberration correction by maximizing generalized sharpness metrics," J. Opt. Soc. Am. A 20, 609-620 (2003).
[CrossRef]

2001 (1)

J.-F. Le Gargasson, M. Glanc, and P. Léna, "Retinal imaging with adaptive optics," C. R. Acad. Sci., Ser IV Phys. Astrophys. 2, 1131-1138 (2001).

1998 (2)

J. C. He, S. Marcos, R. H. Webb, and S. A. Burns, "Measurement of the wave-font aberration of the eye by a fast psychophysical procedure," J. Opt. Soc. Am. A 15, 1-8 (1998).
[CrossRef]

M. A. Losada and R. Navarro, "Point spread function of the human eye obtained by a dual double-pass method," Pure Appl. Opt. 7, L7-L13 (1998).
[CrossRef]

1996 (1)

1994 (1)

1980 (1)

1974 (1)

Adler, J.

S. Zommer, E. N. Ribak, S. G. Lipson, and J. Adler, "Simulated annealing in ocular adaptive optics," Opt. Lett. 31, 1-3 (2006).
[CrossRef]

Bille, J. F.

Bonora, S.

G. Naletto, F. Frassetto, N. Codogno, E. Grisan, S. Bonora, V. Da Deppo, and A. Ruggeri, "No wavefront sensor adaptive optics system for compensation of primary aberrations by software analysis of a point source image. 2. Tests," Appl. Opt. 46, 0000-0000 (2007). [same issue (81590)]
[CrossRef]

Booth, M. J.

Buffington, A.

Burns, S. A.

Carhart, G. W.

Codogno, N.

G. Naletto, F. Frassetto, N. Codogno, E. Grisan, S. Bonora, V. Da Deppo, and A. Ruggeri, "No wavefront sensor adaptive optics system for compensation of primary aberrations by software analysis of a point source image. 2. Tests," Appl. Opt. 46, 0000-0000 (2007). [same issue (81590)]
[CrossRef]

Da Deppo, V.

G. Naletto, F. Frassetto, N. Codogno, E. Grisan, S. Bonora, V. Da Deppo, and A. Ruggeri, "No wavefront sensor adaptive optics system for compensation of primary aberrations by software analysis of a point source image. 2. Tests," Appl. Opt. 46, 0000-0000 (2007). [same issue (81590)]
[CrossRef]

Dainty, J. C.

L. Murray, J. C. Dainty, and E. Daly, "Wavefront correction through image sharpness maximisation," Proc. SPIE 5823, 40-47 (2005).
[CrossRef]

Daly, E.

L. Murray, J. C. Dainty, and E. Daly, "Wavefront correction through image sharpness maximisation," Proc. SPIE 5823, 40-47 (2005).
[CrossRef]

Doble, N.

N. Doble and D. R. Williams, "The application of MEMS technology for adaptive optics in vision science," IEEE J. Sel. Top. Quantum Electron. 10, 629-635 (2004).
[CrossRef]

Fankhauser, F.

F. Fankhauser, P. F. Niederer, S. Kwasniewska, and F. van der Zypen, "Supernormal vision, high-resolution retinal imaging, multiphoton imaging and nanosurgery of the cornea-a review," Technol. Health Care 12, 443-453 (2004).

Fernández, E. J.

Fienup, J. R.

Frassetto, F.

G. Naletto, F. Frassetto, N. Codogno, E. Grisan, S. Bonora, V. Da Deppo, and A. Ruggeri, "No wavefront sensor adaptive optics system for compensation of primary aberrations by software analysis of a point source image. 2. Tests," Appl. Opt. 46, 0000-0000 (2007). [same issue (81590)]
[CrossRef]

Gendron, E.

M. Glanc, E. Gendron, F. Lacombe, D. Lafaille, J. F. Le Gargasson, and P. Léna, "Towards wide-field retinal imaging with adaptive optics," Opt. Commun. 230, 225-238 (2004).
[CrossRef]

Glanc, M.

M. Glanc, E. Gendron, F. Lacombe, D. Lafaille, J. F. Le Gargasson, and P. Léna, "Towards wide-field retinal imaging with adaptive optics," Opt. Commun. 230, 225-238 (2004).
[CrossRef]

J.-F. Le Gargasson, M. Glanc, and P. Léna, "Retinal imaging with adaptive optics," C. R. Acad. Sci., Ser IV Phys. Astrophys. 2, 1131-1138 (2001).

Goelz, S.

Grimm, B.

Grisan, E.

G. Naletto, F. Frassetto, N. Codogno, E. Grisan, S. Bonora, V. Da Deppo, and A. Ruggeri, "No wavefront sensor adaptive optics system for compensation of primary aberrations by software analysis of a point source image. 2. Tests," Appl. Opt. 46, 0000-0000 (2007). [same issue (81590)]
[CrossRef]

He, J. C.

Kwasniewska, S.

F. Fankhauser, P. F. Niederer, S. Kwasniewska, and F. van der Zypen, "Supernormal vision, high-resolution retinal imaging, multiphoton imaging and nanosurgery of the cornea-a review," Technol. Health Care 12, 443-453 (2004).

Lacombe, F.

M. Glanc, E. Gendron, F. Lacombe, D. Lafaille, J. F. Le Gargasson, and P. Léna, "Towards wide-field retinal imaging with adaptive optics," Opt. Commun. 230, 225-238 (2004).
[CrossRef]

Lafaille, D.

M. Glanc, E. Gendron, F. Lacombe, D. Lafaille, J. F. Le Gargasson, and P. Léna, "Towards wide-field retinal imaging with adaptive optics," Opt. Commun. 230, 225-238 (2004).
[CrossRef]

Le Gargasson, J. F.

M. Glanc, E. Gendron, F. Lacombe, D. Lafaille, J. F. Le Gargasson, and P. Léna, "Towards wide-field retinal imaging with adaptive optics," Opt. Commun. 230, 225-238 (2004).
[CrossRef]

Le Gargasson, J.-F.

J.-F. Le Gargasson, M. Glanc, and P. Léna, "Retinal imaging with adaptive optics," C. R. Acad. Sci., Ser IV Phys. Astrophys. 2, 1131-1138 (2001).

Léna, P.

M. Glanc, E. Gendron, F. Lacombe, D. Lafaille, J. F. Le Gargasson, and P. Léna, "Towards wide-field retinal imaging with adaptive optics," Opt. Commun. 230, 225-238 (2004).
[CrossRef]

J.-F. Le Gargasson, M. Glanc, and P. Léna, "Retinal imaging with adaptive optics," C. R. Acad. Sci., Ser IV Phys. Astrophys. 2, 1131-1138 (2001).

Liang, J.

Lipson, S. G.

S. Zommer, E. N. Ribak, S. G. Lipson, and J. Adler, "Simulated annealing in ocular adaptive optics," Opt. Lett. 31, 1-3 (2006).
[CrossRef]

Losada, M. A.

M. A. Losada and R. Navarro, "Point spread function of the human eye obtained by a dual double-pass method," Pure Appl. Opt. 7, L7-L13 (1998).
[CrossRef]

Marcos, S.

Miller, J. J.

Mu, G.-G.

W. Quan, Z.-Q. Wang, G.-G. Mu, and L. Ning, "Correction of the aberrations in the human eyes with svag1 thin-film transistor liquid-crystal display," Optik 114, 467-471 (2003).
[CrossRef]

Muller, R. A.

Murray, L.

L. Murray, J. C. Dainty, and E. Daly, "Wavefront correction through image sharpness maximisation," Proc. SPIE 5823, 40-47 (2005).
[CrossRef]

Naletto, G.

G. Naletto, F. Frassetto, N. Codogno, E. Grisan, S. Bonora, V. Da Deppo, and A. Ruggeri, "No wavefront sensor adaptive optics system for compensation of primary aberrations by software analysis of a point source image. 2. Tests," Appl. Opt. 46, 0000-0000 (2007). [same issue (81590)]
[CrossRef]

Navarro, R.

M. A. Losada and R. Navarro, "Point spread function of the human eye obtained by a dual double-pass method," Pure Appl. Opt. 7, L7-L13 (1998).
[CrossRef]

Niederer, P. F.

F. Fankhauser, P. F. Niederer, S. Kwasniewska, and F. van der Zypen, "Supernormal vision, high-resolution retinal imaging, multiphoton imaging and nanosurgery of the cornea-a review," Technol. Health Care 12, 443-453 (2004).

Ning, L.

W. Quan, Z.-Q. Wang, G.-G. Mu, and L. Ning, "Correction of the aberrations in the human eyes with svag1 thin-film transistor liquid-crystal display," Optik 114, 467-471 (2003).
[CrossRef]

Pruidze, D. V.

Quan, W.

W. Quan, Z.-Q. Wang, G.-G. Mu, and L. Ning, "Correction of the aberrations in the human eyes with svag1 thin-film transistor liquid-crystal display," Optik 114, 467-471 (2003).
[CrossRef]

Ribak, E. N.

S. Zommer, E. N. Ribak, S. G. Lipson, and J. Adler, "Simulated annealing in ocular adaptive optics," Opt. Lett. 31, 1-3 (2006).
[CrossRef]

Ricklin, J. C.

Ruggeri, A.

G. Naletto, F. Frassetto, N. Codogno, E. Grisan, S. Bonora, V. Da Deppo, and A. Ruggeri, "No wavefront sensor adaptive optics system for compensation of primary aberrations by software analysis of a point source image. 2. Tests," Appl. Opt. 46, 0000-0000 (2007). [same issue (81590)]
[CrossRef]

Silva, D.

Vabre, L.

van der Zypen, F.

F. Fankhauser, P. F. Niederer, S. Kwasniewska, and F. van der Zypen, "Supernormal vision, high-resolution retinal imaging, multiphoton imaging and nanosurgery of the cornea-a review," Technol. Health Care 12, 443-453 (2004).

Voelz, D. G.

Vorontsov, M. A.

Wang, J.

Wang, Z.-Q.

W. Quan, Z.-Q. Wang, G.-G. Mu, and L. Ning, "Correction of the aberrations in the human eyes with svag1 thin-film transistor liquid-crystal display," Optik 114, 467-471 (2003).
[CrossRef]

Webb, R. H.

Williams, D. R.

N. Doble and D. R. Williams, "The application of MEMS technology for adaptive optics in vision science," IEEE J. Sel. Top. Quantum Electron. 10, 629-635 (2004).
[CrossRef]

Zommer, S.

S. Zommer, E. N. Ribak, S. G. Lipson, and J. Adler, "Simulated annealing in ocular adaptive optics," Opt. Lett. 31, 1-3 (2006).
[CrossRef]

Appl. Opt. (2)

G. Naletto, F. Frassetto, N. Codogno, E. Grisan, S. Bonora, V. Da Deppo, and A. Ruggeri, "No wavefront sensor adaptive optics system for compensation of primary aberrations by software analysis of a point source image. 2. Tests," Appl. Opt. 46, 0000-0000 (2007). [same issue (81590)]
[CrossRef]

J. Wang and D. Silva, "Wave-front interpretation with Zernike polynomials," Appl. Opt. 19, 1510-1518 (1980).
[CrossRef] [PubMed]

C. R. Acad. Sci., Ser IV Phys. Astrophys. (1)

J.-F. Le Gargasson, M. Glanc, and P. Léna, "Retinal imaging with adaptive optics," C. R. Acad. Sci., Ser IV Phys. Astrophys. 2, 1131-1138 (2001).

IEEE J. Sel. Top. Quantum Electron. (1)

N. Doble and D. R. Williams, "The application of MEMS technology for adaptive optics in vision science," IEEE J. Sel. Top. Quantum Electron. 10, 629-635 (2004).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (4)

Opt. Commun. (1)

M. Glanc, E. Gendron, F. Lacombe, D. Lafaille, J. F. Le Gargasson, and P. Léna, "Towards wide-field retinal imaging with adaptive optics," Opt. Commun. 230, 225-238 (2004).
[CrossRef]

Opt. Express (2)

Opt. Lett. (2)

M. J. Booth, "Wavefront sensorless adaptive optics for large aberrations," Opt. Lett. 32, 5-7 (2007).
[CrossRef]

S. Zommer, E. N. Ribak, S. G. Lipson, and J. Adler, "Simulated annealing in ocular adaptive optics," Opt. Lett. 31, 1-3 (2006).
[CrossRef]

Optik (1)

W. Quan, Z.-Q. Wang, G.-G. Mu, and L. Ning, "Correction of the aberrations in the human eyes with svag1 thin-film transistor liquid-crystal display," Optik 114, 467-471 (2003).
[CrossRef]

Proc. SPIE (1)

L. Murray, J. C. Dainty, and E. Daly, "Wavefront correction through image sharpness maximisation," Proc. SPIE 5823, 40-47 (2005).
[CrossRef]

Pure Appl. Opt. (1)

M. A. Losada and R. Navarro, "Point spread function of the human eye obtained by a dual double-pass method," Pure Appl. Opt. 7, L7-L13 (1998).
[CrossRef]

Technol. Health Care (1)

F. Fankhauser, P. F. Niederer, S. Kwasniewska, and F. van der Zypen, "Supernormal vision, high-resolution retinal imaging, multiphoton imaging and nanosurgery of the cornea-a review," Technol. Health Care 12, 443-453 (2004).

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Figures (10)

Fig. 1
Fig. 1

Typical image of a point source image affected by coma.

Fig. 2
Fig. 2

Point source image affected by astigmatism at different focal plane positions. The position (b) corresponds to the so-called “circle of least confusion.”

Fig. 3
Fig. 3

Flow chart of the proposed algorithm. Zernike coefficients related to each aberration are hierarchically and separately estimated.

Fig. 4
Fig. 4

(Color online) Coma detection: the white star is c 2 and the white circle is c 1 , as defined in the text.

Fig. 5
Fig. 5

Equivalent model of the simple optical system simulated for testing the aberration correction algorithm.

Fig. 6
Fig. 6

Comparison plots of the Zernike coefficients of the fixed surface Σ 1 versus the estimated coefficient describing the surface Σ 2 . In Fig. 7(a) the plot for the z 9 coefficient is represented, Fig. 7(b) for the z 7 coefficient, Fig. 7(c) for the z 8 coefficient, Fig. 7(d) for the z 4 coefficient, Fig. 7(e) for the z 6 coefficient, and finally Fig. 7(f) for the z 5 coefficient.

Fig. 7
Fig. 7

Progressive aberration estimation and correction. The xy plot scale is the same for all the images, however the color map is different, so that each image is normalized to its maximum intensity to visualize the point source shape. (a) Original aberrated point image. Maximum intensity is 2 × 10 4 , and the function f ( c) described in [10] is 0.19. (b) Aberrated point image after spherical correction. Maximum intensity is 5 × 10 4 , and the function f(c) described in [10] is 0.99. (c) Aberrated point image after coma correction. Maximum intensity is 6 × 10 4 , and the function f(c) described in [10] is 0.98. (d) Aberrated point image after defocus correction. Maximum intensity is 16 × 10 4 , and the function f(c) described in [10] is 0.90. (e) Aberrated point image after astigmatism correction. Maximum intensity is 16 × 10 4 , and the function f(c) described in [10] is 0.90.

Fig. 8
Fig. 8

Error distribution in the estimation of the Zernike coefficients. Each box has lines at the lower quartile, median, and upper quartile values. The whiskers are lines extending to the most extreme value of the data inside 1.5 times the interquartile range. Outliers are data with values beyond the ends of the whiskers, and are plotted with a single cross.

Fig. 9
Fig. 9

Error distribution in the estimation of the Zernike coefficients obtained with the direct maximization proposed in [10]. Each box has lines at the lower quartile, median, and upper quartile values. The whiskers are lines extending to the most extreme value of the data inside 1.5 times the interquartile range. Outliers are data with values beyond the ends of the whiskers, and are plotted with a single cross.

Fig. 10
Fig. 10

Number of iterations needed for the proposed algorithm to converge for each single type of aberration to be estimated. Each box has lines at the lower quartile, median, and upper quartile values. The whiskers are lines extending to the most extreme value of the data inside 1.5 times the interquartile range. Outliers are data with values beyond the ends of the whiskers, and are plotted with a single cross.

Tables (2)

Tables Icon

Table 1 Absolute Error on the Estimates of the Zernike Coefficients of the Aberrating Surface

Tables Icon

Table 2 Absolute Error on the Estimates of the Zernike Coefficients of the Aberrating Surface Obtained with the Direct Maximization Method Proposed in [10]

Equations (16)

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

S 1 = I ( x , y ) d x d y .
J s = { S 1 for   stage   1 μ I for   stage   2 .
z 9 , Σ 2 ( i ) = z 9 , Σ 2 ( i 1 ) + δ z ( i ) ,
J s ( i 2 ) J s ( i 1 ) δ z ( i ) = δ z ( i 1 ) ,
J s ( i 2 ) < J s ( i 1 ) δ z ( i ) = 0.5 δ z ( i 1 ) .
J c = ( 1 I 2 ( x , y ) d x d y ) c 1 c 2 2 ,
z 7 , Σ 2 ( i ) = z 7 , Σ 2 ( i 1 ) + δ z ( i ) cos ( ϑ coma ( i ) ) ,
z 8 , Σ 2 ( i ) = z 8 , Σ 2 ( i 1 ) + δ z ( i ) sin ( ϑ coma ( i ) ) .
J c ( i 2 ) > J c ( i 1 ) δ z ( i ) = δ z ( i 1 ) ,
J c ( i 2 ) J c ( i 1 ) δ z ( i ) = 0.5 δ z ( i 1 ) .
S 5 = A I ( x , y ) d x d y ,
J s = { 1 S 5 for   stage   1 max ( Cov ( A ) ) for   stage   2 ,
z 4 , Σ 2 ( i ) = z 4 , Σ 2 ( i 1 ) + δ z ( i ) ,
J d ( i 2 ) > J d ( i 1 ) δ z ( i ) = δ z ( i ) ,
J d ( i 2 ) J d ( i 1 ) δ z ( i ) = 0.5 δ z ( i ) .
ε i = | z i + z ^ i | .

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