Abstract

A feature-based phase retrieval wavefront sensing approach using machine learning is proposed in contrast to the conventional intensity-based approaches. Specifically, the Tchebichef moments which are orthogonal in the discrete domain of the image coordinate space are introduced to represent the features of the point spread functions (PSFs) at the in-focus and defocus image planes. The back-propagation artificial neural network, which is one of most wide applied machine learning tool, is utilized to establish the nonlinear mapping between the Tchebichef moment features and the corresponding aberration coefficients of the optical system. The Tchebichef moments can effectively characterize the intensity distribution of the PSFs. Once well trained, the neural network can directly output the aberration coefficients of the optical system to a good precision with these image features serving as the input. Adequate experiments are implemented to demonstrate the effectiveness and accuracy of proposed approach. This work presents a feasible and easy-implemented way to improve the efficiency and robustness of the phase retrieval wavefront sensing.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  25. R. Mukundan, “Some computational aspects of discrete orthonormal moments,” IEEE Trans. Image Process. 13(8), 1055–1059 (2004).
    [Crossref] [PubMed]
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    [Crossref]
  27. A. T. C. Goh, “Back-propagation neural networks for modeling complex systems,” Artif. Intell. Eng. 9(3), 143–151 (1995).
    [Crossref]

2018 (1)

2016 (1)

2014 (1)

L. Deng, “A tutorial survey of architectures, algorithms, and applications for deep learning,” APSIPA Trans. Signal. Inf. Process. 3, e2 (2014).
[Crossref]

2010 (1)

2006 (1)

B. H. Dean, D. L. Aronstein, J. S. Smith, R. Shiri, and D. S. Acton, “Phase Retrieval Algorithm for JWST Flight and Testbed Telescope,” Proc. SPIE 6265, 626511 (2006).
[Crossref]

2004 (1)

R. Mukundan, “Some computational aspects of discrete orthonormal moments,” IEEE Trans. Image Process. 13(8), 1055–1059 (2004).
[Crossref] [PubMed]

2001 (1)

R. Mukundan, S. H. Ong, and P. A. Lee, “Image analysis by Tchebichef moments,” IEEE Trans. Image Process. 10(9), 1357–1364 (2001).
[Crossref] [PubMed]

1996 (1)

A. K. Jain, J. Mao, and K. M. Mohiuddin, “Artificial neural networks: A tutorial,” Computer 29(3), 31–44 (1996).
[Crossref]

1995 (2)

A. T. C. Goh, “Back-propagation neural networks for modeling complex systems,” Artif. Intell. Eng. 9(3), 143–151 (1995).
[Crossref]

P. Groot, “Phase-shift calibration errors in interferometers with spherical Fizeau cavities,” Appl. Opt. 34(16), 2856–2863 (1995).
[Crossref] [PubMed]

1993 (6)

1992 (1)

1991 (1)

D. G. Sandler, T. K. Barrett, D. A. Palmer, R. Q. Fugate, and W. J. Wild, “Use of a neural network to control an adaptive optics system for an astronomical telescope,” Nature 351(6324), 300–302 (1991).
[Crossref]

1990 (1)

J. R. P. Angel, P. Wizinowich, M. Lloyd-Hart, and D. Sandler, “Adaptive optics for array telescopes using neural-network techniques,” Nature 348(6298), 221–224 (1990).
[Crossref]

1982 (2)

R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21(5), 829–832 (1982).
[Crossref]

J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21(15), 2758–2769 (1982).
[Crossref] [PubMed]

1979 (1)

R. A. Gonsalves and R. C. Hidlaw, “Wavefront sensing by phase retrieval,” Proc. SPIE 207, 32–39 (1979).
[Crossref]

1973 (1)

D. L. Misell, “An examination of an iterative method for the solution of the phase problem in optics and electron optics,” J. Phys. D 6(18), 2200–2225 (1973).
[Crossref]

1972 (1)

R. W. Gerchberg and W. O. Saxton, “A Practical Algorithm for the Determination of Phase from Image and Diffraction Plane Pictures,” Optik (Stuttg.) 35, 237–246 (1972).

1961 (1)

L. D. Harmon, “Studies with artificial neurons. I. Properties and functions of an artificial neuron,” Kybernetik 1(3), 89–101 (1961).
[Crossref] [PubMed]

Acton, D. S.

B. H. Dean, D. L. Aronstein, J. S. Smith, R. Shiri, and D. S. Acton, “Phase Retrieval Algorithm for JWST Flight and Testbed Telescope,” Proc. SPIE 6265, 626511 (2006).
[Crossref]

Angel, J. R. P.

J. R. P. Angel, P. Wizinowich, M. Lloyd-Hart, and D. Sandler, “Adaptive optics for array telescopes using neural-network techniques,” Nature 348(6298), 221–224 (1990).
[Crossref]

Aronstein, D. L.

B. H. Dean, D. L. Aronstein, J. S. Smith, R. Shiri, and D. S. Acton, “Phase Retrieval Algorithm for JWST Flight and Testbed Telescope,” Proc. SPIE 6265, 626511 (2006).
[Crossref]

Barrett, T. K.

T. K. Barrett and D. G. Sandler, “Artificial neural network for the determination of Hubble Space Telescope aberration from stellar images,” Appl. Opt. 32(10), 1720–1727 (1993).
[Crossref] [PubMed]

D. G. Sandler, T. K. Barrett, D. A. Palmer, R. Q. Fugate, and W. J. Wild, “Use of a neural network to control an adaptive optics system for an astronomical telescope,” Nature 351(6324), 300–302 (1991).
[Crossref]

Belenguer, T.

Dass, S. C.

Dean, B. H.

B. H. Dean, D. L. Aronstein, J. S. Smith, R. Shiri, and D. S. Acton, “Phase Retrieval Algorithm for JWST Flight and Testbed Telescope,” Proc. SPIE 6265, 626511 (2006).
[Crossref]

Deng, L.

L. Deng, “A tutorial survey of architectures, algorithms, and applications for deep learning,” APSIPA Trans. Signal. Inf. Process. 3, e2 (2014).
[Crossref]

Dumont, P.

Fienup, J. R.

Fugate, R. Q.

D. G. Sandler, T. K. Barrett, D. A. Palmer, R. Q. Fugate, and W. J. Wild, “Use of a neural network to control an adaptive optics system for an astronomical telescope,” Nature 351(6324), 300–302 (1991).
[Crossref]

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, “A Practical Algorithm for the Determination of Phase from Image and Diffraction Plane Pictures,” Optik (Stuttg.) 35, 237–246 (1972).

Goh, A. T. C.

A. T. C. Goh, “Back-propagation neural networks for modeling complex systems,” Artif. Intell. Eng. 9(3), 143–151 (1995).
[Crossref]

Gonsalves, R. A.

R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21(5), 829–832 (1982).
[Crossref]

R. A. Gonsalves and R. C. Hidlaw, “Wavefront sensing by phase retrieval,” Proc. SPIE 207, 32–39 (1979).
[Crossref]

González-Fernandez, L.

Groot, P.

Harmon, L. D.

L. D. Harmon, “Studies with artificial neurons. I. Properties and functions of an artificial neuron,” Kybernetik 1(3), 89–101 (1961).
[Crossref] [PubMed]

Hidlaw, R. C.

R. A. Gonsalves and R. C. Hidlaw, “Wavefront sensing by phase retrieval,” Proc. SPIE 207, 32–39 (1979).
[Crossref]

Hinton, G. E.

A. Krizhevsky, I. Sutskever, and G. E. Hinton, “Imagenet classification with deep convolutional neural networks,” in Advances in Neural Information Processing Systems (2012), pp. 1097–1105.

Jain, A. K.

A. K. Jain, J. Mao, and K. M. Mohiuddin, “Artificial neural networks: A tutorial,” Computer 29(3), 31–44 (1996).
[Crossref]

Krizhevsky, A.

A. Krizhevsky, I. Sutskever, and G. E. Hinton, “Imagenet classification with deep convolutional neural networks,” in Advances in Neural Information Processing Systems (2012), pp. 1097–1105.

Kumar, A.

Lee, P. A.

R. Mukundan, S. H. Ong, and P. A. Lee, “Image analysis by Tchebichef moments,” IEEE Trans. Image Process. 10(9), 1357–1364 (2001).
[Crossref] [PubMed]

Lim, C.-L.

Lloyd-Hart, M.

J. R. P. Angel, P. Wizinowich, M. Lloyd-Hart, and D. Sandler, “Adaptive optics for array telescopes using neural-network techniques,” Nature 348(6298), 221–224 (1990).
[Crossref]

Mao, J.

A. K. Jain, J. Mao, and K. M. Mohiuddin, “Artificial neural networks: A tutorial,” Computer 29(3), 31–44 (1996).
[Crossref]

Marron, J. C.

Misell, D. L.

D. L. Misell, “An examination of an iterative method for the solution of the phase problem in optics and electron optics,” J. Phys. D 6(18), 2200–2225 (1973).
[Crossref]

Mohiuddin, K. M.

A. K. Jain, J. Mao, and K. M. Mohiuddin, “Artificial neural networks: A tutorial,” Computer 29(3), 31–44 (1996).
[Crossref]

Mukundan, R.

R. Mukundan, “Some computational aspects of discrete orthonormal moments,” IEEE Trans. Image Process. 13(8), 1055–1059 (2004).
[Crossref] [PubMed]

R. Mukundan, S. H. Ong, and P. A. Lee, “Image analysis by Tchebichef moments,” IEEE Trans. Image Process. 10(9), 1357–1364 (2001).
[Crossref] [PubMed]

Ong, S. H.

R. Mukundan, S. H. Ong, and P. A. Lee, “Image analysis by Tchebichef moments,” IEEE Trans. Image Process. 10(9), 1357–1364 (2001).
[Crossref] [PubMed]

Paine, S. W.

Palmer, D. A.

D. G. Sandler, T. K. Barrett, D. A. Palmer, R. Q. Fugate, and W. J. Wild, “Use of a neural network to control an adaptive optics system for an astronomical telescope,” Nature 351(6324), 300–302 (1991).
[Crossref]

Paramesran, R.

Paxman, R. G.

Quiroga, J. A.

Redding, D.

Roddier, C.

Roddier, F.

Sandler, D.

J. R. P. Angel, P. Wizinowich, M. Lloyd-Hart, and D. Sandler, “Adaptive optics for array telescopes using neural-network techniques,” Nature 348(6298), 221–224 (1990).
[Crossref]

Sandler, D. G.

T. K. Barrett and D. G. Sandler, “Artificial neural network for the determination of Hubble Space Telescope aberration from stellar images,” Appl. Opt. 32(10), 1720–1727 (1993).
[Crossref] [PubMed]

D. G. Sandler, T. K. Barrett, D. A. Palmer, R. Q. Fugate, and W. J. Wild, “Use of a neural network to control an adaptive optics system for an astronomical telescope,” Nature 351(6324), 300–302 (1991).
[Crossref]

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, “A Practical Algorithm for the Determination of Phase from Image and Diffraction Plane Pictures,” Optik (Stuttg.) 35, 237–246 (1972).

Schulz, T. J.

Seldin, J. H.

Shiri, R.

B. H. Dean, D. L. Aronstein, J. S. Smith, R. Shiri, and D. S. Acton, “Phase Retrieval Algorithm for JWST Flight and Testbed Telescope,” Proc. SPIE 6265, 626511 (2006).
[Crossref]

Smith, J. S.

B. H. Dean, D. L. Aronstein, J. S. Smith, R. Shiri, and D. S. Acton, “Phase Retrieval Algorithm for JWST Flight and Testbed Telescope,” Proc. SPIE 6265, 626511 (2006).
[Crossref]

Sutskever, I.

A. Krizhevsky, I. Sutskever, and G. E. Hinton, “Imagenet classification with deep convolutional neural networks,” in Advances in Neural Information Processing Systems (2012), pp. 1097–1105.

Vargas, J.

Wild, W. J.

D. G. Sandler, T. K. Barrett, D. A. Palmer, R. Q. Fugate, and W. J. Wild, “Use of a neural network to control an adaptive optics system for an astronomical telescope,” Nature 351(6324), 300–302 (1991).
[Crossref]

Wizinowich, P.

J. R. P. Angel, P. Wizinowich, M. Lloyd-Hart, and D. Sandler, “Adaptive optics for array telescopes using neural-network techniques,” Nature 348(6298), 221–224 (1990).
[Crossref]

Yu, J.

Appl. Opt. (8)

APSIPA Trans. Signal. Inf. Process. (1)

L. Deng, “A tutorial survey of architectures, algorithms, and applications for deep learning,” APSIPA Trans. Signal. Inf. Process. 3, e2 (2014).
[Crossref]

Artif. Intell. Eng. (1)

A. T. C. Goh, “Back-propagation neural networks for modeling complex systems,” Artif. Intell. Eng. 9(3), 143–151 (1995).
[Crossref]

Computer (1)

A. K. Jain, J. Mao, and K. M. Mohiuddin, “Artificial neural networks: A tutorial,” Computer 29(3), 31–44 (1996).
[Crossref]

IEEE Trans. Image Process. (2)

R. Mukundan, S. H. Ong, and P. A. Lee, “Image analysis by Tchebichef moments,” IEEE Trans. Image Process. 10(9), 1357–1364 (2001).
[Crossref] [PubMed]

R. Mukundan, “Some computational aspects of discrete orthonormal moments,” IEEE Trans. Image Process. 13(8), 1055–1059 (2004).
[Crossref] [PubMed]

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

J. Phys. D (1)

D. L. Misell, “An examination of an iterative method for the solution of the phase problem in optics and electron optics,” J. Phys. D 6(18), 2200–2225 (1973).
[Crossref]

Kybernetik (1)

L. D. Harmon, “Studies with artificial neurons. I. Properties and functions of an artificial neuron,” Kybernetik 1(3), 89–101 (1961).
[Crossref] [PubMed]

Nature (2)

D. G. Sandler, T. K. Barrett, D. A. Palmer, R. Q. Fugate, and W. J. Wild, “Use of a neural network to control an adaptive optics system for an astronomical telescope,” Nature 351(6324), 300–302 (1991).
[Crossref]

J. R. P. Angel, P. Wizinowich, M. Lloyd-Hart, and D. Sandler, “Adaptive optics for array telescopes using neural-network techniques,” Nature 348(6298), 221–224 (1990).
[Crossref]

Opt. Eng. (1)

R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21(5), 829–832 (1982).
[Crossref]

Opt. Lett. (2)

Optik (Stuttg.) (1)

R. W. Gerchberg and W. O. Saxton, “A Practical Algorithm for the Determination of Phase from Image and Diffraction Plane Pictures,” Optik (Stuttg.) 35, 237–246 (1972).

Proc. SPIE (2)

B. H. Dean, D. L. Aronstein, J. S. Smith, R. Shiri, and D. S. Acton, “Phase Retrieval Algorithm for JWST Flight and Testbed Telescope,” Proc. SPIE 6265, 626511 (2006).
[Crossref]

R. A. Gonsalves and R. C. Hidlaw, “Wavefront sensing by phase retrieval,” Proc. SPIE 207, 32–39 (1979).
[Crossref]

Other (2)

A. Krizhevsky, I. Sutskever, and G. E. Hinton, “Imagenet classification with deep convolutional neural networks,” in Advances in Neural Information Processing Systems (2012), pp. 1097–1105.

I. Sutskever, J. Martens, G. Dahl, and G. Hinton, “On the importance of initialization and momentum in deep learning,” International Conference on International Conference on Machine Learning 38, III-1139 (2013).

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Figures (8)

Fig. 1
Fig. 1 The mathematical model of an artificial neuron.
Fig. 2
Fig. 2 Back-propagation algorithm for a neural network with three layers.
Fig. 3
Fig. 3 Sketch map of the feature-based phase retrieval wavefront sensing approach using machine learning.
Fig. 4
Fig. 4 Application procedure of the feature-based phase retrieval wavefront sensing approach using machine learning.
Fig. 5
Fig. 5 The sketch (a) and physical map (b) of the optical system used in the experiment.
Fig. 6
Fig. 6 Distribution of the error between the targets and the actual outputs of the network in the form of histogram in the presence of image noise and the error in the defocus distance.
Fig. 7
Fig. 7 Comparison between the 20 pairs of real PSF images and the 20 pairs of regenerated PSF images. In each of the 20 images in this figure, the upper two PSFs are collected form the optical system (at different focal planes) and the two PSF images below are generated with the recovered aberration coefficients. We can recognize that the regenerated PSF images bear strong similarities with those real collected, which qualitatively demonstrate the accuracy of the recovered aberration coefficients.
Fig. 8
Fig. 8 Distribution of the error between the targets and the actual outputs of the network in the form of histogram for the case shown in Table 3.

Tables (5)

Tables Icon

Table 1 Ranges of different aberration coefficients for generating the data set

Tables Icon

Table 2 Comparison between the astigmatic (C5/C6) and coma (C7/C8) aberration coefficients measured by interferometer (A) and those recovered using the proposed approach (B)

Tables Icon

Table 3 Another case with a different range of aberration coefficients

Tables Icon

Table 4 Demonstration for accuracy of the new neural network with a smaller capture range

Tables Icon

Table 5 The results of applying the new neural network to some cases beyond its capture range

Equations (10)

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

y k = φ ( i = 1 m x i w i k b k ) ,
φ ( z ) = 1 1 + exp ( z ) .
x = 0 N 1 t m ( x ) t n ( x ) = ρ ( n , N ) δ m n ,
f ( x , y ) = m = 0 N 1 n = 0 N 1 T m n t m ( x ) t n ( y ) ,
T p q = 1 ρ ( p , N ) ρ ( p , N ) x = 0 N 1 y = 0 N 1 t p ( x ) t q ( y ) f ( x , y ) ,
t n ( x ) = n ! k = 0 n ( 1 ) n k ( N 1 k n k ) ( n + k n ) ( x k ) .
ρ ( n , N ) = N ( N 2 1 ) ( N 2 2 2 ) ( N 2 n 2 ) 2 n + 1 .
t ˜ n ( x ) = 2 n 1 n t ˜ 1 ( x ) t ˜ n 1 ( x ) n 1 n [ 1 ( n 1 ) 2 N 2 ] t ˜ n 2 ( x ) ,
t ˜ 0 ( x ) = 1 ,
t ˜ 1 ( x ) = ( 2 x + 1 N ) / N .

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