Abstract

To optimally compensate for time-varying phase aberrations with adaptive optics, a model of the dynamics of the aberrations is required to predict the phase aberration at the next time step. We model the time-varying behavior of a phase aberration, expressed in Zernike modes, by assuming that the temporal dynamics of the Zernike coefficients can be described by a vector-valued autoregressive (VAR) model. We propose an iterative method based on a convex heuristic for a rank-constrained optimization problem, to jointly estimate the parameters of the VAR model and the Zernike coefficients from a time series of measurements of the point-spread function (PSF) of the optical system. By assuming the phase aberration is small, the relation between aberration and PSF measurements can be approximated by a quadratic function. As such, our method is a blind identification method for linear dynamics in a stochastic Wiener system with a quadratic nonlinearity at the output and a phase retrieval method that uses a time-evolution-model constraint and a single image at every time step.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2019 (1)

B. Sinquin and M. Verhaegen, “Quarks: identification of large-scale Kronecker vector-autoregressive models,” IEEE Trans. Autom. Control 64, 448–463 (2019).
[Crossref]

2018 (1)

G. Monchen, B. Sinquin, and M. Verhaegen, “Recursive Kronecker-based vector autoregressive identification for large-scale adaptive optics,” IEEE Trans. Control Syst. Technol. 99, 1–8 (2018).
[Crossref]

2014 (2)

R. Wallin and A. Hansson, “Maximum likelihood estimation of linear SISO models subject to missing output data and missing input data,” Int. J. Control 87, 2358–2364 (2014).
[Crossref]

Y. N. Sulai and A. Dubra, “Non-common path aberration correction in an adaptive optics scanning ophthalmoscope,” Biomed. Opt. Express 5, 3059–3073 (2014).
[Crossref]

2013 (4)

C. Smith, R. Marinică, A. Den Dekker, M. Verhaegen, V. Korkiakoski, C. Keller, and N. Doelman, “Iterative linear focal-plane wavefront correction,” J. Opt. Soc. Am. A 30, 2002–2011 (2013).
[Crossref]

Y. Hu, D. Zhang, J. Ye, X. Li, and X. He, “Fast and accurate matrix completion via truncated nuclear norm regularization,” IEEE Trans. Pattern Anal. Mach. Intell. 35, 2117–2130 (2013).
[Crossref]

A. Wills, T. B. Schön, L. Ljung, and B. Ninness, “Identification of Hammerstein–Wiener models,” Automatica 49, 70–81 (2013).
[Crossref]

F. Lindsten, T. B. Schön, and M. I. Jordan, “Bayesian semiparametric Wiener system identification,” Automatica 49, 2053–2063(2013).
[Crossref]

2011 (1)

2009 (1)

L. Vanbeylen, R. Pintelon, and J. Schoukens, “Blind maximum-likelihood identification of Wiener systems,” IEEE Trans. Signal Process. 57, 3017–3029 (2009).
[Crossref]

2008 (2)

2007 (1)

2005 (1)

M. Van Noort, L. R. Van Der Voort, and M. G. Löfdahl, “Solar image restoration by use of multi-frame blind de-convolution with multiple objects and phase diversity,” Solar Phys. 228, 191–215 (2005).
[Crossref]

2004 (2)

2003 (2)

M. Hartung, A. Blanc, T. Fusco, F. Lacombe, L. Mugnier, G. Rousset, and R. Lenzen, “Calibration of NAOS and CONICA static aberrations—experimental results,” Astron. Astrophys. 399, 385–394(2003).
[Crossref]

A. Blanc, T. Fusco, M. Hartung, L. Mugnier, and G. Rousset, “Calibration of NAOS and CONICA static aberrations—application of the phase diversity technique,” Astron. Astrophys. 399, 373–383 (2003).
[Crossref]

2002 (1)

2001 (2)

T. Van Gestel, J. A. Suykens, P. Van Dooren, and B. De Moor, “Identification of stable models in subspace identification by using regularization,” IEEE Trans. Autom. Control 46, 1416–1420(2001).
[Crossref]

B. C. Platt and R. Shack, “History and principles of Shack–Hartmann wavefront sensing,” J. Refractive Surg. 17, S573–S577 (2001).
[Crossref]

1996 (1)

D. Acton, D. Soltau, and W. Schmidt, “Full-field wavefront measurements with phase diversity,” Astron. Astrophys. 309, 661–672 (1996).

1982 (2)

R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21, 215829 (1982).
[Crossref]

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

Acton, D.

D. Acton, D. Soltau, and W. Schmidt, “Full-field wavefront measurements with phase diversity,” Astron. Astrophys. 309, 661–672 (1996).

Balakrishnan, V. R.

L. Vandenberghe, V. R. Balakrishnan, R. Wallin, A. Hansson, and T. Roh, “Interior-point algorithms for semidefinite programming problems derived from the KYP lemma,” in Positive Polynomials in Control (Springer, 2005), pp. 195–238.

Beuzit, J. L.

T. Fusco, C. Petit, G. Rousset, J. F. Sauvage, A. Blanc, J. M. Conan, and J. L. Beuzit, “Optimization of the pre-compensation of non-common path aberrations for adaptive optics systems,” in Adaptive Optics: Methods, Analysis and Applications (Optical Society of America, 2005), paper AWB2.

Blanc, A.

M. Hartung, A. Blanc, T. Fusco, F. Lacombe, L. Mugnier, G. Rousset, and R. Lenzen, “Calibration of NAOS and CONICA static aberrations—experimental results,” Astron. Astrophys. 399, 385–394(2003).
[Crossref]

A. Blanc, T. Fusco, M. Hartung, L. Mugnier, and G. Rousset, “Calibration of NAOS and CONICA static aberrations—application of the phase diversity technique,” Astron. Astrophys. 399, 373–383 (2003).
[Crossref]

T. Fusco, C. Petit, G. Rousset, J. F. Sauvage, A. Blanc, J. M. Conan, and J. L. Beuzit, “Optimization of the pre-compensation of non-common path aberrations for adaptive optics systems,” in Adaptive Optics: Methods, Analysis and Applications (Optical Society of America, 2005), paper AWB2.

Campbell, M. C.

Conan, J. M.

T. Fusco, C. Petit, G. Rousset, J. F. Sauvage, A. Blanc, J. M. Conan, and J. L. Beuzit, “Optimization of the pre-compensation of non-common path aberrations for adaptive optics systems,” in Adaptive Optics: Methods, Analysis and Applications (Optical Society of America, 2005), paper AWB2.

Conan, J.-M.

B. Le Roux, J.-M. Conan, C. Kulcsár, H.-F. Raynaud, L. M. Mugnier, and T. Fusco, “Optimal control law for classical and multiconjugate adaptive optics,” J. Opt. Soc. Am. A 21, 1261–1276 (2004).
[Crossref]

C. Kulcsár, H.-F. Raynaud, J.-M. Conan, C. Correia, and C. Petit, “Control design and turbulent phase models in adaptive optics: a state-space interpretation,” in Adaptive Optics: Methods, Analysis and Applications (Optical Society of America, 2009), paper AOWB1.

Cornia, A.

Correia, C.

C. Kulcsár, H.-F. Raynaud, J.-M. Conan, C. Correia, and C. Petit, “Control design and turbulent phase models in adaptive optics: a state-space interpretation,” in Adaptive Optics: Methods, Analysis and Applications (Optical Society of America, 2009), paper AOWB1.

Dandy, S.

De Moor, B.

T. Van Gestel, J. A. Suykens, P. Van Dooren, and B. De Moor, “Identification of stable models in subspace identification by using regularization,” IEEE Trans. Autom. Control 46, 1416–1420(2001).
[Crossref]

Den Dekker, A.

Doelman, N.

C. Smith, R. Marinică, A. Den Dekker, M. Verhaegen, V. Korkiakoski, C. Keller, and N. Doelman, “Iterative linear focal-plane wavefront correction,” J. Opt. Soc. Am. A 30, 2002–2011 (2013).
[Crossref]

K. Hinnen, M. Verhaegen, and N. Doelman, “A data-driven H2-optimal control approach for adaptive optics,” IEEE Trans. Control Syst. Technol. 16, 381–395 (2008).
[Crossref]

Doelman, R.

R. Doelman and M. Verhaegen, “Sequential convex relaxation for convex optimization with bilinear matrix equalities,” in Proceedings of the European Control Conference (2016).

Donnelly, W. J.

Dubra, A.

Fienup, J. R.

Fusco, T.

L. Mugnier, J.-F. Sauvage, T. Fusco, A. Cornia, and S. Dandy, “On-line long-exposure phase diversity: a powerful tool for sensing quasi-static aberrations of extreme adaptive optics imaging systems,” Opt. Express 16, 18406–18416 (2008).
[Crossref]

J.-F. Sauvage, T. Fusco, G. Rousset, and C. Petit, “Calibration and precompensation of noncommon path aberrations for extreme adaptive optics,” J. Opt. Soc. Am. A 24, 2334–2346 (2007).
[Crossref]

B. Le Roux, J.-M. Conan, C. Kulcsár, H.-F. Raynaud, L. M. Mugnier, and T. Fusco, “Optimal control law for classical and multiconjugate adaptive optics,” J. Opt. Soc. Am. A 21, 1261–1276 (2004).
[Crossref]

A. Blanc, T. Fusco, M. Hartung, L. Mugnier, and G. Rousset, “Calibration of NAOS and CONICA static aberrations—application of the phase diversity technique,” Astron. Astrophys. 399, 373–383 (2003).
[Crossref]

M. Hartung, A. Blanc, T. Fusco, F. Lacombe, L. Mugnier, G. Rousset, and R. Lenzen, “Calibration of NAOS and CONICA static aberrations—experimental results,” Astron. Astrophys. 399, 385–394(2003).
[Crossref]

T. Fusco, C. Petit, G. Rousset, J. F. Sauvage, A. Blanc, J. M. Conan, and J. L. Beuzit, “Optimization of the pre-compensation of non-common path aberrations for adaptive optics systems,” in Adaptive Optics: Methods, Analysis and Applications (Optical Society of America, 2005), paper AWB2.

Gonsalves, R. A.

R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21, 215829 (1982).
[Crossref]

Hansson, A.

R. Wallin and A. Hansson, “Maximum likelihood estimation of linear SISO models subject to missing output data and missing input data,” Int. J. Control 87, 2358–2364 (2014).
[Crossref]

L. Vandenberghe, V. R. Balakrishnan, R. Wallin, A. Hansson, and T. Roh, “Interior-point algorithms for semidefinite programming problems derived from the KYP lemma,” in Positive Polynomials in Control (Springer, 2005), pp. 195–238.

Hartung, M.

A. Blanc, T. Fusco, M. Hartung, L. Mugnier, and G. Rousset, “Calibration of NAOS and CONICA static aberrations—application of the phase diversity technique,” Astron. Astrophys. 399, 373–383 (2003).
[Crossref]

M. Hartung, A. Blanc, T. Fusco, F. Lacombe, L. Mugnier, G. Rousset, and R. Lenzen, “Calibration of NAOS and CONICA static aberrations—experimental results,” Astron. Astrophys. 399, 385–394(2003).
[Crossref]

Haverbeke, N.

N. Haverbeke, “Efficient numerical methods for moving horizon,” dissertation (Katholieke Universiteit Leuven, 2011).

He, X.

Y. Hu, D. Zhang, J. Ye, X. Li, and X. He, “Fast and accurate matrix completion via truncated nuclear norm regularization,” IEEE Trans. Pattern Anal. Mach. Intell. 35, 2117–2130 (2013).
[Crossref]

Hebert, T. J.

Hinnen, K.

K. Hinnen, M. Verhaegen, and N. Doelman, “A data-driven H2-optimal control approach for adaptive optics,” IEEE Trans. Control Syst. Technol. 16, 381–395 (2008).
[Crossref]

Hofer, H.

Hu, Y.

Y. Hu, D. Zhang, J. Ye, X. Li, and X. He, “Fast and accurate matrix completion via truncated nuclear norm regularization,” IEEE Trans. Pattern Anal. Mach. Intell. 35, 2117–2130 (2013).
[Crossref]

Jordan, M. I.

F. Lindsten, T. B. Schön, and M. I. Jordan, “Bayesian semiparametric Wiener system identification,” Automatica 49, 2053–2063(2013).
[Crossref]

F. Lindsten, T. Schön, and M. I. Jordan, A Semiparametric Bayesian Approach to Wiener System Identification (Linköping University Electronic, 2011).

Keller, C.

Korkiakoski, V.

Kulcsár, C.

B. Le Roux, J.-M. Conan, C. Kulcsár, H.-F. Raynaud, L. M. Mugnier, and T. Fusco, “Optimal control law for classical and multiconjugate adaptive optics,” J. Opt. Soc. Am. A 21, 1261–1276 (2004).
[Crossref]

C. Kulcsár, H.-F. Raynaud, J.-M. Conan, C. Correia, and C. Petit, “Control design and turbulent phase models in adaptive optics: a state-space interpretation,” in Adaptive Optics: Methods, Analysis and Applications (Optical Society of America, 2009), paper AOWB1.

Lacombe, F.

M. Hartung, A. Blanc, T. Fusco, F. Lacombe, L. Mugnier, G. Rousset, and R. Lenzen, “Calibration of NAOS and CONICA static aberrations—experimental results,” Astron. Astrophys. 399, 385–394(2003).
[Crossref]

Le Mignant, D.

Le Roux, B.

Lenzen, R.

M. Hartung, A. Blanc, T. Fusco, F. Lacombe, L. Mugnier, G. Rousset, and R. Lenzen, “Calibration of NAOS and CONICA static aberrations—experimental results,” Astron. Astrophys. 399, 385–394(2003).
[Crossref]

Li, C.

Li, X.

Y. Hu, D. Zhang, J. Ye, X. Li, and X. He, “Fast and accurate matrix completion via truncated nuclear norm regularization,” IEEE Trans. Pattern Anal. Mach. Intell. 35, 2117–2130 (2013).
[Crossref]

Lindsten, F.

F. Lindsten, T. B. Schön, and M. I. Jordan, “Bayesian semiparametric Wiener system identification,” Automatica 49, 2053–2063(2013).
[Crossref]

F. Lindsten, T. Schön, and M. I. Jordan, A Semiparametric Bayesian Approach to Wiener System Identification (Linköping University Electronic, 2011).

Ljung, L.

A. Wills, T. B. Schön, L. Ljung, and B. Ninness, “Identification of Hammerstein–Wiener models,” Automatica 49, 70–81 (2013).
[Crossref]

A. Wills, T. B. Schön, L. Ljung, and B. Ninness, “Blind identification of Wiener models,” in Proceedings of the 18th IFAC World Congress, Milan, Italy (2011).

Lofberg, J.

J. Lofberg, “YALMIP: a toolbox for modeling and optimization in MATLAB,” in IEEE International Symposium on Computer Aided Control Systems Design (IEEE, 2004), pp. 284–289.

Lofdahl, M. G.

M. G. Lofdahl, “Multi-frame blind deconvolution with linear equality constraints,” in Image Reconstruction from Incomplete Data II (International Society for Optics and Photonics, 2002), Vol. 4792, pp. 146–156.

Löfdahl, M. G.

M. Van Noort, L. R. Van Der Voort, and M. G. Löfdahl, “Solar image restoration by use of multi-frame blind de-convolution with multiple objects and phase diversity,” Solar Phys. 228, 191–215 (2005).
[Crossref]

Macintosh, B. A.

Marinica, R.

C. Smith, R. Marinică, A. Den Dekker, M. Verhaegen, V. Korkiakoski, C. Keller, and N. Doelman, “Iterative linear focal-plane wavefront correction,” J. Opt. Soc. Am. A 30, 2002–2011 (2013).
[Crossref]

R. Marinică, C. S. Smith, and M. Verhaegen, “State feedback control with quadratic output for wavefront correction in adaptive optics,” in IEEE 52nd Annual Conference on Decision and Control (CDC) (IEEE, 2013), pp. 3475–3480.

Monchen, G.

G. Monchen, B. Sinquin, and M. Verhaegen, “Recursive Kronecker-based vector autoregressive identification for large-scale adaptive optics,” IEEE Trans. Control Syst. Technol. 99, 1–8 (2018).
[Crossref]

Mugnier, L.

L. Mugnier, J.-F. Sauvage, T. Fusco, A. Cornia, and S. Dandy, “On-line long-exposure phase diversity: a powerful tool for sensing quasi-static aberrations of extreme adaptive optics imaging systems,” Opt. Express 16, 18406–18416 (2008).
[Crossref]

A. Blanc, T. Fusco, M. Hartung, L. Mugnier, and G. Rousset, “Calibration of NAOS and CONICA static aberrations—application of the phase diversity technique,” Astron. Astrophys. 399, 373–383 (2003).
[Crossref]

M. Hartung, A. Blanc, T. Fusco, F. Lacombe, L. Mugnier, G. Rousset, and R. Lenzen, “Calibration of NAOS and CONICA static aberrations—experimental results,” Astron. Astrophys. 399, 385–394(2003).
[Crossref]

Mugnier, L. M.

Ninness, B.

A. Wills, T. B. Schön, L. Ljung, and B. Ninness, “Identification of Hammerstein–Wiener models,” Automatica 49, 70–81 (2013).
[Crossref]

A. Wills, T. B. Schön, L. Ljung, and B. Ninness, “Blind identification of Wiener models,” in Proceedings of the 18th IFAC World Congress, Milan, Italy (2011).

Ozbay, H.

O. Toker and H. Ozbay, “On the NP-hardness of solving bilinear matrix inequalities and simultaneous stabilization with static output feedback,” in Proceedings of the American Control Conference (IEEE, 1995), Vol. 4, pp. 2525–2526.

Petit, C.

J.-F. Sauvage, T. Fusco, G. Rousset, and C. Petit, “Calibration and precompensation of noncommon path aberrations for extreme adaptive optics,” J. Opt. Soc. Am. A 24, 2334–2346 (2007).
[Crossref]

T. Fusco, C. Petit, G. Rousset, J. F. Sauvage, A. Blanc, J. M. Conan, and J. L. Beuzit, “Optimization of the pre-compensation of non-common path aberrations for adaptive optics systems,” in Adaptive Optics: Methods, Analysis and Applications (Optical Society of America, 2005), paper AWB2.

C. Kulcsár, H.-F. Raynaud, J.-M. Conan, C. Correia, and C. Petit, “Control design and turbulent phase models in adaptive optics: a state-space interpretation,” in Adaptive Optics: Methods, Analysis and Applications (Optical Society of America, 2009), paper AOWB1.

Pintelon, R.

L. Vanbeylen, R. Pintelon, and J. Schoukens, “Blind maximum-likelihood identification of Wiener systems,” IEEE Trans. Signal Process. 57, 3017–3029 (2009).
[Crossref]

Platt, B. C.

B. C. Platt and R. Shack, “History and principles of Shack–Hartmann wavefront sensing,” J. Refractive Surg. 17, S573–S577 (2001).
[Crossref]

Porter, J.

Queener, H.

Raynaud, H.-F.

B. Le Roux, J.-M. Conan, C. Kulcsár, H.-F. Raynaud, L. M. Mugnier, and T. Fusco, “Optimal control law for classical and multiconjugate adaptive optics,” J. Opt. Soc. Am. A 21, 1261–1276 (2004).
[Crossref]

C. Kulcsár, H.-F. Raynaud, J.-M. Conan, C. Correia, and C. Petit, “Control design and turbulent phase models in adaptive optics: a state-space interpretation,” in Adaptive Optics: Methods, Analysis and Applications (Optical Society of America, 2009), paper AOWB1.

Roh, T.

L. Vandenberghe, V. R. Balakrishnan, R. Wallin, A. Hansson, and T. Roh, “Interior-point algorithms for semidefinite programming problems derived from the KYP lemma,” in Positive Polynomials in Control (Springer, 2005), pp. 195–238.

Romero-Borja, F.

Roorda, A.

Rousset, G.

J.-F. Sauvage, T. Fusco, G. Rousset, and C. Petit, “Calibration and precompensation of noncommon path aberrations for extreme adaptive optics,” J. Opt. Soc. Am. A 24, 2334–2346 (2007).
[Crossref]

M. Hartung, A. Blanc, T. Fusco, F. Lacombe, L. Mugnier, G. Rousset, and R. Lenzen, “Calibration of NAOS and CONICA static aberrations—experimental results,” Astron. Astrophys. 399, 385–394(2003).
[Crossref]

A. Blanc, T. Fusco, M. Hartung, L. Mugnier, and G. Rousset, “Calibration of NAOS and CONICA static aberrations—application of the phase diversity technique,” Astron. Astrophys. 399, 373–383 (2003).
[Crossref]

T. Fusco, C. Petit, G. Rousset, J. F. Sauvage, A. Blanc, J. M. Conan, and J. L. Beuzit, “Optimization of the pre-compensation of non-common path aberrations for adaptive optics systems,” in Adaptive Optics: Methods, Analysis and Applications (Optical Society of America, 2005), paper AWB2.

Sauvage, J. F.

T. Fusco, C. Petit, G. Rousset, J. F. Sauvage, A. Blanc, J. M. Conan, and J. L. Beuzit, “Optimization of the pre-compensation of non-common path aberrations for adaptive optics systems,” in Adaptive Optics: Methods, Analysis and Applications (Optical Society of America, 2005), paper AWB2.

Sauvage, J.-F.

Schmidt, W.

D. Acton, D. Soltau, and W. Schmidt, “Full-field wavefront measurements with phase diversity,” Astron. Astrophys. 309, 661–672 (1996).

Schön, T.

F. Lindsten, T. Schön, and M. I. Jordan, A Semiparametric Bayesian Approach to Wiener System Identification (Linköping University Electronic, 2011).

Schön, T. B.

A. Wills, T. B. Schön, L. Ljung, and B. Ninness, “Identification of Hammerstein–Wiener models,” Automatica 49, 70–81 (2013).
[Crossref]

F. Lindsten, T. B. Schön, and M. I. Jordan, “Bayesian semiparametric Wiener system identification,” Automatica 49, 2053–2063(2013).
[Crossref]

A. Wills, T. B. Schön, L. Ljung, and B. Ninness, “Blind identification of Wiener models,” in Proceedings of the 18th IFAC World Congress, Milan, Italy (2011).

Schoukens, J.

L. Vanbeylen, R. Pintelon, and J. Schoukens, “Blind maximum-likelihood identification of Wiener systems,” IEEE Trans. Signal Process. 57, 3017–3029 (2009).
[Crossref]

Shack, R.

B. C. Platt and R. Shack, “History and principles of Shack–Hartmann wavefront sensing,” J. Refractive Surg. 17, S573–S577 (2001).
[Crossref]

Sinquin, B.

B. Sinquin and M. Verhaegen, “Quarks: identification of large-scale Kronecker vector-autoregressive models,” IEEE Trans. Autom. Control 64, 448–463 (2019).
[Crossref]

G. Monchen, B. Sinquin, and M. Verhaegen, “Recursive Kronecker-based vector autoregressive identification for large-scale adaptive optics,” IEEE Trans. Control Syst. Technol. 99, 1–8 (2018).
[Crossref]

Smith, C.

Smith, C. S.

R. Marinică, C. S. Smith, and M. Verhaegen, “State feedback control with quadratic output for wavefront correction in adaptive optics,” in IEEE 52nd Annual Conference on Decision and Control (CDC) (IEEE, 2013), pp. 3475–3480.

Soltau, D.

D. Acton, D. Soltau, and W. Schmidt, “Full-field wavefront measurements with phase diversity,” Astron. Astrophys. 309, 661–672 (1996).

Sredar, N.

Sulai, Y. N.

Suykens, J. A.

T. Van Gestel, J. A. Suykens, P. Van Dooren, and B. De Moor, “Identification of stable models in subspace identification by using regularization,” IEEE Trans. Autom. Control 46, 1416–1420(2001).
[Crossref]

Toker, O.

O. Toker and H. Ozbay, “On the NP-hardness of solving bilinear matrix inequalities and simultaneous stabilization with static output feedback,” in Proceedings of the American Control Conference (IEEE, 1995), Vol. 4, pp. 2525–2526.

van Dam, M. A.

Van Der Voort, L. R.

M. Van Noort, L. R. Van Der Voort, and M. G. Löfdahl, “Solar image restoration by use of multi-frame blind de-convolution with multiple objects and phase diversity,” Solar Phys. 228, 191–215 (2005).
[Crossref]

Van Dooren, P.

T. Van Gestel, J. A. Suykens, P. Van Dooren, and B. De Moor, “Identification of stable models in subspace identification by using regularization,” IEEE Trans. Autom. Control 46, 1416–1420(2001).
[Crossref]

Van Gestel, T.

T. Van Gestel, J. A. Suykens, P. Van Dooren, and B. De Moor, “Identification of stable models in subspace identification by using regularization,” IEEE Trans. Autom. Control 46, 1416–1420(2001).
[Crossref]

Van Noort, M.

M. Van Noort, L. R. Van Der Voort, and M. G. Löfdahl, “Solar image restoration by use of multi-frame blind de-convolution with multiple objects and phase diversity,” Solar Phys. 228, 191–215 (2005).
[Crossref]

Vanbeylen, L.

L. Vanbeylen, R. Pintelon, and J. Schoukens, “Blind maximum-likelihood identification of Wiener systems,” IEEE Trans. Signal Process. 57, 3017–3029 (2009).
[Crossref]

Vandenberghe, L.

L. Vandenberghe, V. R. Balakrishnan, R. Wallin, A. Hansson, and T. Roh, “Interior-point algorithms for semidefinite programming problems derived from the KYP lemma,” in Positive Polynomials in Control (Springer, 2005), pp. 195–238.

Verdult, V.

M. Verhaegen and V. Verdult, Filtering and System Identification: a Least Squares Approach (Cambridge University, 2007).

Verhaegen, M.

B. Sinquin and M. Verhaegen, “Quarks: identification of large-scale Kronecker vector-autoregressive models,” IEEE Trans. Autom. Control 64, 448–463 (2019).
[Crossref]

G. Monchen, B. Sinquin, and M. Verhaegen, “Recursive Kronecker-based vector autoregressive identification for large-scale adaptive optics,” IEEE Trans. Control Syst. Technol. 99, 1–8 (2018).
[Crossref]

C. Smith, R. Marinică, A. Den Dekker, M. Verhaegen, V. Korkiakoski, C. Keller, and N. Doelman, “Iterative linear focal-plane wavefront correction,” J. Opt. Soc. Am. A 30, 2002–2011 (2013).
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K. Hinnen, M. Verhaegen, and N. Doelman, “A data-driven H2-optimal control approach for adaptive optics,” IEEE Trans. Control Syst. Technol. 16, 381–395 (2008).
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R. Marinică, C. S. Smith, and M. Verhaegen, “State feedback control with quadratic output for wavefront correction in adaptive optics,” in IEEE 52nd Annual Conference on Decision and Control (CDC) (IEEE, 2013), pp. 3475–3480.

M. Verhaegen and V. Verdult, Filtering and System Identification: a Least Squares Approach (Cambridge University, 2007).

R. Doelman and M. Verhaegen, “Sequential convex relaxation for convex optimization with bilinear matrix equalities,” in Proceedings of the European Control Conference (2016).

Wallin, R.

R. Wallin and A. Hansson, “Maximum likelihood estimation of linear SISO models subject to missing output data and missing input data,” Int. J. Control 87, 2358–2364 (2014).
[Crossref]

L. Vandenberghe, V. R. Balakrishnan, R. Wallin, A. Hansson, and T. Roh, “Interior-point algorithms for semidefinite programming problems derived from the KYP lemma,” in Positive Polynomials in Control (Springer, 2005), pp. 195–238.

Wills, A.

A. Wills, T. B. Schön, L. Ljung, and B. Ninness, “Identification of Hammerstein–Wiener models,” Automatica 49, 70–81 (2013).
[Crossref]

A. Wills, T. B. Schön, L. Ljung, and B. Ninness, “Blind identification of Wiener models,” in Proceedings of the 18th IFAC World Congress, Milan, Italy (2011).

Ye, J.

Y. Hu, D. Zhang, J. Ye, X. Li, and X. He, “Fast and accurate matrix completion via truncated nuclear norm regularization,” IEEE Trans. Pattern Anal. Mach. Intell. 35, 2117–2130 (2013).
[Crossref]

Zhang, D.

Y. Hu, D. Zhang, J. Ye, X. Li, and X. He, “Fast and accurate matrix completion via truncated nuclear norm regularization,” IEEE Trans. Pattern Anal. Mach. Intell. 35, 2117–2130 (2013).
[Crossref]

Appl. Opt. (2)

Astron. Astrophys. (3)

D. Acton, D. Soltau, and W. Schmidt, “Full-field wavefront measurements with phase diversity,” Astron. Astrophys. 309, 661–672 (1996).

M. Hartung, A. Blanc, T. Fusco, F. Lacombe, L. Mugnier, G. Rousset, and R. Lenzen, “Calibration of NAOS and CONICA static aberrations—experimental results,” Astron. Astrophys. 399, 385–394(2003).
[Crossref]

A. Blanc, T. Fusco, M. Hartung, L. Mugnier, and G. Rousset, “Calibration of NAOS and CONICA static aberrations—application of the phase diversity technique,” Astron. Astrophys. 399, 373–383 (2003).
[Crossref]

Automatica (2)

A. Wills, T. B. Schön, L. Ljung, and B. Ninness, “Identification of Hammerstein–Wiener models,” Automatica 49, 70–81 (2013).
[Crossref]

F. Lindsten, T. B. Schön, and M. I. Jordan, “Bayesian semiparametric Wiener system identification,” Automatica 49, 2053–2063(2013).
[Crossref]

Biomed. Opt. Express (1)

IEEE Trans. Autom. Control (2)

B. Sinquin and M. Verhaegen, “Quarks: identification of large-scale Kronecker vector-autoregressive models,” IEEE Trans. Autom. Control 64, 448–463 (2019).
[Crossref]

T. Van Gestel, J. A. Suykens, P. Van Dooren, and B. De Moor, “Identification of stable models in subspace identification by using regularization,” IEEE Trans. Autom. Control 46, 1416–1420(2001).
[Crossref]

IEEE Trans. Control Syst. Technol. (2)

G. Monchen, B. Sinquin, and M. Verhaegen, “Recursive Kronecker-based vector autoregressive identification for large-scale adaptive optics,” IEEE Trans. Control Syst. Technol. 99, 1–8 (2018).
[Crossref]

K. Hinnen, M. Verhaegen, and N. Doelman, “A data-driven H2-optimal control approach for adaptive optics,” IEEE Trans. Control Syst. Technol. 16, 381–395 (2008).
[Crossref]

IEEE Trans. Pattern Anal. Mach. Intell. (1)

Y. Hu, D. Zhang, J. Ye, X. Li, and X. He, “Fast and accurate matrix completion via truncated nuclear norm regularization,” IEEE Trans. Pattern Anal. Mach. Intell. 35, 2117–2130 (2013).
[Crossref]

IEEE Trans. Signal Process. (1)

L. Vanbeylen, R. Pintelon, and J. Schoukens, “Blind maximum-likelihood identification of Wiener systems,” IEEE Trans. Signal Process. 57, 3017–3029 (2009).
[Crossref]

Int. J. Control (1)

R. Wallin and A. Hansson, “Maximum likelihood estimation of linear SISO models subject to missing output data and missing input data,” Int. J. Control 87, 2358–2364 (2014).
[Crossref]

J. Opt. Soc. Am. A (3)

J. Refractive Surg. (1)

B. C. Platt and R. Shack, “History and principles of Shack–Hartmann wavefront sensing,” J. Refractive Surg. 17, S573–S577 (2001).
[Crossref]

Opt. Eng. (1)

R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21, 215829 (1982).
[Crossref]

Opt. Express (3)

Solar Phys. (1)

M. Van Noort, L. R. Van Der Voort, and M. G. Löfdahl, “Solar image restoration by use of multi-frame blind de-convolution with multiple objects and phase diversity,” Solar Phys. 228, 191–215 (2005).
[Crossref]

Other (13)

M. G. Lofdahl, “Multi-frame blind deconvolution with linear equality constraints,” in Image Reconstruction from Incomplete Data II (International Society for Optics and Photonics, 2002), Vol. 4792, pp. 146–156.

M. Verhaegen and V. Verdult, Filtering and System Identification: a Least Squares Approach (Cambridge University, 2007).

C. Kulcsár, H.-F. Raynaud, J.-M. Conan, C. Correia, and C. Petit, “Control design and turbulent phase models in adaptive optics: a state-space interpretation,” in Adaptive Optics: Methods, Analysis and Applications (Optical Society of America, 2009), paper AOWB1.

N. Haverbeke, “Efficient numerical methods for moving horizon,” dissertation (Katholieke Universiteit Leuven, 2011).

R. Doelman and M. Verhaegen, “Sequential convex relaxation for convex optimization with bilinear matrix equalities,” in Proceedings of the European Control Conference (2016).

O. Toker and H. Ozbay, “On the NP-hardness of solving bilinear matrix inequalities and simultaneous stabilization with static output feedback,” in Proceedings of the American Control Conference (IEEE, 1995), Vol. 4, pp. 2525–2526.

J. Lofberg, “YALMIP: a toolbox for modeling and optimization in MATLAB,” in IEEE International Symposium on Computer Aided Control Systems Design (IEEE, 2004), pp. 284–289.

M. Grant and S. Boyd, “CVX: MATLAB software for disciplined convex programming, version 2.1,” 2014, http://cvxr.com/cvx .

F. Lindsten, T. Schön, and M. I. Jordan, A Semiparametric Bayesian Approach to Wiener System Identification (Linköping University Electronic, 2011).

R. Marinică, C. S. Smith, and M. Verhaegen, “State feedback control with quadratic output for wavefront correction in adaptive optics,” in IEEE 52nd Annual Conference on Decision and Control (CDC) (IEEE, 2013), pp. 3475–3480.

T. Fusco, C. Petit, G. Rousset, J. F. Sauvage, A. Blanc, J. M. Conan, and J. L. Beuzit, “Optimization of the pre-compensation of non-common path aberrations for adaptive optics systems,” in Adaptive Optics: Methods, Analysis and Applications (Optical Society of America, 2005), paper AWB2.

A. Wills, T. B. Schön, L. Ljung, and B. Ninness, “Blind identification of Wiener models,” in Proceedings of the 18th IFAC World Congress, Milan, Italy (2011).

L. Vandenberghe, V. R. Balakrishnan, R. Wallin, A. Hansson, and T. Roh, “Interior-point algorithms for semidefinite programming problems derived from the KYP lemma,” in Positive Polynomials in Control (Springer, 2005), pp. 195–238.

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Figures (7)

Fig. 1.
Fig. 1. Classical optical setup for estimating the temporal dynamics of an aberrated wavefront. This setup includes three lenses with focal length f .
Fig. 2.
Fig. 2. Overview of identification methods. y denotes the measurement, ϕ is the phase aberration, α is the vector of Zernike coefficients, w k a noise signal, k is a time index, and f ( · ) is the model function. I. Phase retrieval first, then model estimation. II. Model estimate based on phase measurements. III. Our method: model estimation and phase retrieval from PSF measurements.
Fig. 3.
Fig. 3. On the left is the PSF at a time step of k = 100 . Outlined in red are the 25 pixel values used in the identification. On the right, top, are the Zernike coefficients for an example data set for k = 1 , , 100 . At the bottom are the time series for k = 1 , , 100 for the corresponding pixel values.
Fig. 4.
Fig. 4. VAF of the estimated state sequences for the first experiment. T-S is for the initial guess from the two-step least squares and pr.m. is for the initial guess from the solution of the proposed method. Note that from this figure, it is apparent that the first least squares problem of the two-step least squares solution (36) fails to produce a good estimate of the states.
Fig. 5.
Fig. 5. VAF of the estimated state sequences for the second experiment. T-S is for the initial guess from the two-step least squares and pr.m. is for the initial guess from the solution of the proposed method.
Fig. 6.
Fig. 6. Comparison of RMS error for the next predicted state by the proposed method, states remain constant, and the true model for the first experiment.
Fig. 7.
Fig. 7. Comparison of RMS for the next predicted state by the proposed method, states remain constant, and the true model for the first experiment.

Tables (2)

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Algorithm 1. Sequential convex optimization-based identification

Tables Icon

Table 1. Settings for the Two Numerical Experiments

Equations (45)

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VAF ( α , α ^ ) = max ( 0 , ( 1 k = 1 K α k α ^ k 2 2 k = 1 K α k 2 2 ) ) ,
n = 1 , 2 X n blkdiag ( X 1 , X 2 ) = ( X 1 0 0 X 2 ) .
P ( x , y ) = A ( x , y ) exp ( j ϕ ( x , y ) ) ,
ϕ ¯ a ( x , y , α ) r = 1 s Z r ( x , y ) α r ,
ϕ ¯ ( x , y , α , β ) = ϕ ¯ a ( x , y , α ) + ϕ ¯ d ( x , y , β ) ,
ϕ ¯ ( x , y , α , β ) = ϕ ¯ ( x , y , α + β , 0 ) ,
ϕ ( α , β ) = [ ϕ ¯ ( x 1 , y 1 , α , β ) ϕ ¯ ( x m , y 1 , α , β ) ϕ ¯ ( x 1 , y m , α , β ) ϕ ¯ ( x m , y m , α , β ) ] ,
vec ( ϕ ( α , β ) ) = Z ( α + β ) ,
Γ = [ A ( x 1 , y 1 ) A ( x m , y 1 ) A ( x 1 , y m ) A ( x m , y m ) ] .
y ( α , β ) = vec ( | F { Γ exp ( j vec 1 ( Z ( α + β ) ) ) } | 2 ) ,
y i ( α , β ) = D 0 , i ( β ) + D 1 , i ( β ) α + O ( α 2 ) ,
D 0 , i ( β ) = y i ( α , β ) | α = 0 R , D 1 , i ( β ) = y i ( α , β ) α | α = 0 R 1 × s .
y i ( α , β ) = D 0 , i ( β ) + D 1 , i ( β ) α + 1 2 α T D 2 , i ( β ) α + O ( α 3 ) ,
D 2 , i ( β ) = 2 y i ( α , β ) α T α | α = 0 R s × s .
VAF ( y i ( α , β ) D 0 , i ( β ) , D 1 , i ( β ) α + 1 2 α T D 2 , i ( β ) α ) ) > 0.9 ,
vec ( ϕ ( α k , β k ) ) = Z ( α k + β k ) ,
α k = f ( α k 1 , , α k N , w k ) = A 1 α k 1 + + A N α k N + w k ,
α k = f ( α k 1 , , α N , w k ) = A 1 α k 1 + + A N α k N + w k , y i ( α k , β k ) = D 0 , i ( β k ) + D 1 , i ( β k ) α k + α k T D 2 , i ( β k ) α k + v i , k ,
minimize A k , α k , w k , v i , k i = 1 K w k 2 2 + γ k = 1 K i = 1 p 2 v i , k 2 2 subject to α k = f ( α k 1 , , α N , w k ) , y i ( α k , β k ) = D 0 , i ( β k ) + D 1 , i ( β k ) α k + α k T D 2 , i ( β k ) α k + v i , k ,
( α K α N + 1 ) A K = ( A 1 A N ) A ( α K 1 α K 2 α N α K 2 α K 3 α N 1 α K N α 1 ) H + W ,
y i ( α k , β k ) D 0 , i ( β k ) D 1 , i ( β k ) α k v i , k = α k T D 2 , i ( β k ) α k .
D y = i , k y i ( α k , β k ) D 0 , i ( β k ) D 1 , i ( β k ) α k , D α = i , k α k , D 2 = i , k D 2 , i ( β k ) , V ^ = i , k v i , k .
minimize α , A , W ^ , V ^ W ^ F 2 + γ V ^ F 2 subject to A K W ^ = A H D y V ^ = D α T D 2 D α .
L ( A , P , B , C , X , Y ) = ( C + A P Y + X P B + X P Y ( A + X ) P P ( B + Y ) P ) .
rank L ( A , P , B , C , X , Y ) = rank P A P B = C
M VAR L ( A , I N s , H , A K W ^ , X 1 , Y 1 ) , M meas L ( D α T , D 2 , D α , D y V ^ , X 2 , Y 2 ) .
rank M VAR = rank I N s = N s , rank M meas = rank D 2
minimize α , A , W ^ , V ^ W ^ F 2 + γ V ^ F 2 subject to rank M VAR = N s rank M meas = rank D 2 .
minimize A , α , W ^ , V ^ W ^ F 2 + γ V ^ F 2 + ξ ( λ M VAR * + M meas * ) .
Q VAR L ( A , I N s , H , A K , X 1 , Y 1 ) , Q meas L ( D α T , D 2 , D α , D y , X 2 , Y 2 ) .
minimize A , α λ Q VAR * + Q meas * .
W ^ * A K * A * H * , V ^ * D y ( D α * ) T D 2 D α * .
X 1 + = A * , Y 1 + = A K * , X 2 + = ( D α * ) T , Y 2 + = D α * .
ϕ ¯ ( α k , β k ) = Z 2 2 α k ( 1 ) + Z 3 1 α k ( 2 ) + Z 3 1 α k ( 3 ) + Z 2 0 β k .
β k = { 0.5 k = 1 , 11 , 21 , 0 otherwise ( Experiment 1 ) .
β k = 0.5 k ( Experiment 2 ) .
A s = ( A 1 true A 2 true I 0 ) ,
I ) α ^ = arg min α i , k y i ( α k , β k ) D 0 , i ( β k ) D 1 , i ( β k ) α k F 2 , II ) ( A ^ 1 , A ^ 2 ) = arg min A 1 , A 2 k α ^ k A 1 α ^ k 1 A 2 α ^ k 2 F 2 .
minimize A , α G ( α ) A ¯ + h ( α ) 2 2 ;
minimize A , α G A ¯ + h 2 2 = minimize α ( I G G ) h 2 2 = minimize α P G h 2 2 ,
r α = P G α h + P G h α
r α P G ( G α + h α ) ,
e k = α k A ^ 1 α k 1 A ^ 2 α k 2 .
G = ( I n α K 1 T I n α K 2 T I n α K M T I n α K N T I n α 1 T 0 0 0 0 ) ,
h = ( α K α N + 1 y 1 , K D 0 , 1 ( β k ) D 1 , 1 ( β k ) α K α K T D 2 , 1 ( β k ) α K y p 2 , K D 0 , p 2 ( β k ) D 1 , p 2 ( β k ) α K α K T D 2 , p 2 ( β k ) α K y 1 , 1 D 0 , 1 ( β k ) D 1 , 1 ( β k ) α K α K T D 2 , 1 ( β k ) α K y p 2 , 1 D 0 , p 2 ( β k ) D 1 , p 2 ( β k ) α 1 α 1 T D 2 , p 2 ( β k ) α 1 ) .

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