Abstract

A novel closed-loop control technique for adaptive optics (AO) systems based on a wavelet-based phase reconstruction technique and a woofer–tweeter controller is presented. The wavelet-based reconstruction technique is based on obtaining a Haar decomposition of the phase screen directly from gradient measurements and has been extended here with the use of a Poisson solver to improve performance. This method is O(N) (i.e., a linear computation cost as number of actuators increases) and is the fastest of the known O(N) reconstruction techniques. The controller configuration is based on the woofer–tweeter controller to control low- and high-spatial-frequency aberrations, respectively. The separation of the woofer and tweeter signals is done using a computationally efficient method that is based on the availability of a low-spatial-resolution reconstruction during the wavelet synthesis process. The performance of the proposed technique is evaluated using a simulated AO system and phase screens generated to reflect atmospheric turbulence with various dynamic characteristics. Results indicate that the combination of the wavelet-based phase reconstruction and woofer–tweeter controller leads to very good results with respect to speed and accuracy.

© 2010 Optical Society of America

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References

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  8. L. Gilles, B. Ellerbroek, and C. Vogel, “multigrid preconditioned conjugate-gradient method for large-scale wave-front reconstruction,” J. Opt. Soc. Am. A 19, 1817–1822 (2002).
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    [PubMed]
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    [CrossRef]
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    [CrossRef]
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2009 (1)

2008 (4)

2007 (3)

L. A. Poyneer, B. A. Macintosh, and J.-P. Véran, “Fourier-transform wavefront control with adaptive prediction of the atmosphere,” J. Opt. Soc. Am. A 24, 2645–2660 (2007).
[CrossRef]

R. Conan, C. Bradley, P. Hampton, O. Keskin, A. Hilton, and C. Blain, “Distributed modal command for a two-deformable-mirror adaptive optics system,” Appl. Opt. 46, 4329–4340 (2007).
[CrossRef] [PubMed]

J.-F. Lavigne, J.-P. Véran, and L. Poyneer, “Woofer–tweeter control algorithm for the Gemini planet imager,” in Adaptive Optics: Analysis and Methods; Computational Optical Sensing and Imaging; Digital Holography and Three-Dimensional Imaging; and Signal Recovery and Synthesis (Optical Society of America, 2007), paper AWB5.
[PubMed]

2006 (5)

2005 (1)

2004 (2)

R. G. Dekany, M. C. Britton, D. T. Gavel, B. L. Ellerbroek, G. Herriot, C. E. Max, and J.-P. Véran, “Adaptive optics requirements for TMT,” Proc. SPIE 5490879–890 (2004).
[CrossRef]

M. A. van Dam, D. Le Mignant, and B. A. Macintosh, “Performance of the Keck observatory adaptive-optics system,” Appl. Opt. 43, 5458–5467 (2004).
[CrossRef] [PubMed]

2003 (1)

2002 (5)

L. A. Poyneer, D. T. Gavel, and J. M. Brase, “Fast wavefront reconstruction in large adaptive optics systems with the Fourier transform,” J. Opt. Soc. Am. A 18, 2100–2111 (2002).
[CrossRef]

B. L. Ellerbroek, “Efficient computation of minimum-variance wavefront reconstructors with sparse matrix techniques,” J. Opt. Soc. Am. A 19, 1803–1816 (2002).
[CrossRef]

L. Gilles, B. Ellerbroek, and C. Vogel, “multigrid preconditioned conjugate-gradient method for large-scale wave-front reconstruction,” J. Opt. Soc. Am. A 19, 1817–1822 (2002).
[CrossRef]

D. S. Watkins, Fundamentals of Matrix Computations (Wiley, 2002).
[CrossRef]

J. D. Barchers, “Closed-loop stable control of two deformable mirrors for compensation of amplitude and phase fluctuations,” J. Opt. Soc. Am. 19, 926–945 (2002).
[CrossRef]

2001 (1)

D. L. Fried, “Adaptive optics wave function reconstruction and phase unwrapping when branch points are present,” Opt. Commun. 200, 43–72 (2001).
[CrossRef]

2000 (2)

J.-P. Véran and G. Herriot, “Centroid gain compensation in Shack–Hartmann adaptive optics systems with natural or laser guide star,” J. Opt. Soc. Am. A 17, 1430–1439 (2000).
[CrossRef]

F. Martin and R. Conan, “Optical parameters relevant for high angular resolution at Paranal from GSM instrument and surface layer contribution,” Astrophys. Suppl. Ser. 144, 39–44 (2000).
[CrossRef]

1999 (1)

F. Roddier, Imaging through the Atmosphere (Cambridge University Press, 1999).

1997 (1)

1995 (1)

M. Vetterli and J. Kovacevic, Wavelets and Subband Coding (Prentice-Hall, 1995), Chap. 6.

1988 (2)

T. Nakajima, “Signal-to-noise ratio of the bispectral analysis of speckle interferometry,” J. Opt. Soc. Am. A 5, 1477–1491 (1988).
[CrossRef]

R. T. Frankot and R. Chellappa, “A method for enforcing integrability in shape from shading,” IEEE Trans. Pattern Anal. Mach. Intell. 10, pp. 439–451 (1988).
[CrossRef]

1986 (1)

1980 (1)

J. Herrmann, “Least-squaress wavefront errors of minimum norm,” J. Opt. Soc. Am. 67, 28–35 (1980).
[CrossRef]

1977 (2)

1976 (1)

1965 (1)

R. W. Hockney, “A fast direct solution of Poisson’s equation using Fourier analysis,” J. Assoc. Comput. Mach. 12, 95–113 (1965).
[CrossRef]

Agathoklis, P.

P. J. Hampton, P. Agathoklis, and C. Bradley, “wavefront reconstruction over a circular aperture using gradient data extrapolated via the mirror equations,” Appl. Opt. 48, 4018–4030 (2009).
[CrossRef] [PubMed]

P. J. Hampton, R. Conan, O. Keskin, C. Bradley, and P. Agathoklis, “Self characterization of linear and non-linear adaptive optics systems,” Appl. Opt. 47, 126–134 (2008).
[CrossRef] [PubMed]

P. J. Hampton, P. Agathoklis, and C. Bradley, “A new wave-front reconstruction method for adaptive optics systems using wavelets,” IEEE J. Sel. Top. Signal Process. 2, 781–792 (2008).
[CrossRef]

P. J. Hampton, R. Conan, C. Bradley, and P. Agathoklis, “Control of a woofer tweeter system of deformable mirrors,” Proc. SPIE 6274, 62741Z (2006).
[CrossRef]

Barchers, J. D.

J. D. Barchers, “Closed-loop stable control of two deformable mirrors for compensation of amplitude and phase fluctuations,” J. Opt. Soc. Am. 19, 926–945 (2002).
[CrossRef]

Bierden, P. A.

S. A. Cornelissen, P. A. Bierden, and T. G. Bifano, “Development of a 4096 element deformable mirror for high contrast astronomical imaging,” Proc. SPIE 6306, 630606 (2006).
[CrossRef]

Bifano, T. G.

S. A. Cornelissen, P. A. Bierden, and T. G. Bifano, “Development of a 4096 element deformable mirror for high contrast astronomical imaging,” Proc. SPIE 6306, 630606 (2006).
[CrossRef]

Blain, C.

Bradley, C.

Bradley, C. H.

Brase, J. M.

L. A. Poyneer, D. T. Gavel, and J. M. Brase, “Fast wavefront reconstruction in large adaptive optics systems with the Fourier transform,” J. Opt. Soc. Am. A 18, 2100–2111 (2002).
[CrossRef]

Britton, M. C.

R. G. Dekany, M. C. Britton, D. T. Gavel, B. L. Ellerbroek, G. Herriot, C. E. Max, and J.-P. Véran, “Adaptive optics requirements for TMT,” Proc. SPIE 5490879–890 (2004).
[CrossRef]

Chellappa, R.

R. T. Frankot and R. Chellappa, “A method for enforcing integrability in shape from shading,” IEEE Trans. Pattern Anal. Mach. Intell. 10, pp. 439–451 (1988).
[CrossRef]

Conan, R.

Cornelissen, S. A.

S. A. Cornelissen, P. A. Bierden, and T. G. Bifano, “Development of a 4096 element deformable mirror for high contrast astronomical imaging,” Proc. SPIE 6306, 630606 (2006).
[CrossRef]

Dekany, R. G.

R. G. Dekany, M. C. Britton, D. T. Gavel, B. L. Ellerbroek, G. Herriot, C. E. Max, and J.-P. Véran, “Adaptive optics requirements for TMT,” Proc. SPIE 5490879–890 (2004).
[CrossRef]

Duncan, B. D.

Ellerbroek, B.

Ellerbroek, B. L.

Frankot, R. T.

R. T. Frankot and R. Chellappa, “A method for enforcing integrability in shape from shading,” IEEE Trans. Pattern Anal. Mach. Intell. 10, pp. 439–451 (1988).
[CrossRef]

Freischlad, K.

Fried, D. L.

D. L. Fried, “Adaptive optics wave function reconstruction and phase unwrapping when branch points are present,” Opt. Commun. 200, 43–72 (2001).
[CrossRef]

D. L. Fried, “Least-squares fitting a wave-front distortion estimate to an array of phase-difference measurements,” J. Opt. Soc. Am. 67, 370–375 (1977).
[CrossRef]

Gavel, D. T.

R. G. Dekany, M. C. Britton, D. T. Gavel, B. L. Ellerbroek, G. Herriot, C. E. Max, and J.-P. Véran, “Adaptive optics requirements for TMT,” Proc. SPIE 5490879–890 (2004).
[CrossRef]

L. A. Poyneer, D. T. Gavel, and J. M. Brase, “Fast wavefront reconstruction in large adaptive optics systems with the Fourier transform,” J. Opt. Soc. Am. A 18, 2100–2111 (2002).
[CrossRef]

Gilles, L.

Hampton, P.

Hampton, P. J.

Harris, S. R.

Herriot, G.

R. G. Dekany, M. C. Britton, D. T. Gavel, B. L. Ellerbroek, G. Herriot, C. E. Max, and J.-P. Véran, “Adaptive optics requirements for TMT,” Proc. SPIE 5490879–890 (2004).
[CrossRef]

J.-P. Véran and G. Herriot, “Centroid gain compensation in Shack–Hartmann adaptive optics systems with natural or laser guide star,” J. Opt. Soc. Am. A 17, 1430–1439 (2000).
[CrossRef]

Herrmann, J.

J. Herrmann, “Least-squaress wavefront errors of minimum norm,” J. Opt. Soc. Am. 67, 28–35 (1980).
[CrossRef]

Hilton, A.

Hockney, R. W.

R. W. Hockney, “A fast direct solution of Poisson’s equation using Fourier analysis,” J. Assoc. Comput. Mach. 12, 95–113 (1965).
[CrossRef]

Hudgin, R.

Keskin, O.

Koliopoulos, C.

Kovacevic, J.

M. Vetterli and J. Kovacevic, Wavelets and Subband Coding (Prentice-Hall, 1995), Chap. 6.

Lall, S.

Lavigne, J.-F.

J.-F. Lavigne and J.-P. Véran, “woofer-tweeter control in an adaptive optics system using a Fourier reconstructor,” J. Opt. Soc. Am. A 25, 2271–2279 (2008).
[CrossRef]

J.-F. Lavigne, J.-P. Véran, and L. Poyneer, “Woofer–tweeter control algorithm for the Gemini planet imager,” in Adaptive Optics: Analysis and Methods; Computational Optical Sensing and Imaging; Digital Holography and Three-Dimensional Imaging; and Signal Recovery and Synthesis (Optical Society of America, 2007), paper AWB5.
[PubMed]

Le Mignant, D.

Lessard, L.

Macintosh, B. A.

MacMynowski, D.

Martin, F.

F. Martin and R. Conan, “Optical parameters relevant for high angular resolution at Paranal from GSM instrument and surface layer contribution,” Astrophys. Suppl. Ser. 144, 39–44 (2000).
[CrossRef]

Max, C. E.

R. G. Dekany, M. C. Britton, D. T. Gavel, B. L. Ellerbroek, G. Herriot, C. E. Max, and J.-P. Véran, “Adaptive optics requirements for TMT,” Proc. SPIE 5490879–890 (2004).
[CrossRef]

Nakajima, T.

Noll, R. J.

Northcott, M. J.

Poyneer, L.

J.-F. Lavigne, J.-P. Véran, and L. Poyneer, “Woofer–tweeter control algorithm for the Gemini planet imager,” in Adaptive Optics: Analysis and Methods; Computational Optical Sensing and Imaging; Digital Holography and Three-Dimensional Imaging; and Signal Recovery and Synthesis (Optical Society of America, 2007), paper AWB5.
[PubMed]

Poyneer, L. A.

Rigaut, F.

Roddier, F.

F. Roddier, Imaging through the Atmosphere (Cambridge University Press, 1999).

van Dam, M. A.

Véran, J.-P.

J.-F. Lavigne and J.-P. Véran, “woofer-tweeter control in an adaptive optics system using a Fourier reconstructor,” J. Opt. Soc. Am. A 25, 2271–2279 (2008).
[CrossRef]

J.-F. Lavigne, J.-P. Véran, and L. Poyneer, “Woofer–tweeter control algorithm for the Gemini planet imager,” in Adaptive Optics: Analysis and Methods; Computational Optical Sensing and Imaging; Digital Holography and Three-Dimensional Imaging; and Signal Recovery and Synthesis (Optical Society of America, 2007), paper AWB5.
[PubMed]

L. A. Poyneer, B. A. Macintosh, and J.-P. Véran, “Fourier-transform wavefront control with adaptive prediction of the atmosphere,” J. Opt. Soc. Am. A 24, 2645–2660 (2007).
[CrossRef]

L. A. Poyneer and J.-P. Véran, “Optimal modal Fourier transform wave-front control,” J. Opt. Soc. Am. A 22, 1515–1526 (2005).
[CrossRef]

R. G. Dekany, M. C. Britton, D. T. Gavel, B. L. Ellerbroek, G. Herriot, C. E. Max, and J.-P. Véran, “Adaptive optics requirements for TMT,” Proc. SPIE 5490879–890 (2004).
[CrossRef]

J.-P. Véran and G. Herriot, “Centroid gain compensation in Shack–Hartmann adaptive optics systems with natural or laser guide star,” J. Opt. Soc. Am. A 17, 1430–1439 (2000).
[CrossRef]

Vetterli, M.

M. Vetterli and J. Kovacevic, Wavelets and Subband Coding (Prentice-Hall, 1995), Chap. 6.

Vogel, C.

Vogel, C. R.

Wallace, B. P.

Watkins, D. S.

D. S. Watkins, Fundamentals of Matrix Computations (Wiley, 2002).
[CrossRef]

West, M.

Widiker, J. J.

Yang, Q.

Appl. Opt. (7)

Astrophys. Suppl. Ser. (1)

F. Martin and R. Conan, “Optical parameters relevant for high angular resolution at Paranal from GSM instrument and surface layer contribution,” Astrophys. Suppl. Ser. 144, 39–44 (2000).
[CrossRef]

IEEE J. Sel. Top. Signal Process. (1)

P. J. Hampton, P. Agathoklis, and C. Bradley, “A new wave-front reconstruction method for adaptive optics systems using wavelets,” IEEE J. Sel. Top. Signal Process. 2, 781–792 (2008).
[CrossRef]

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

R. T. Frankot and R. Chellappa, “A method for enforcing integrability in shape from shading,” IEEE Trans. Pattern Anal. Mach. Intell. 10, pp. 439–451 (1988).
[CrossRef]

J. Assoc. Comput. Mach. (1)

R. W. Hockney, “A fast direct solution of Poisson’s equation using Fourier analysis,” J. Assoc. Comput. Mach. 12, 95–113 (1965).
[CrossRef]

J. Opt. Soc. Am. (5)

J. Opt. Soc. Am. A (10)

K. Freischlad and C. Koliopoulos, “Modal estimation of a wave-front from difference measurements using the discrete Fourier transform,” J. Opt. Soc. Am. A 3, 1852–1861 (1986).
[CrossRef]

L. A. Poyneer, D. T. Gavel, and J. M. Brase, “Fast wavefront reconstruction in large adaptive optics systems with the Fourier transform,” J. Opt. Soc. Am. A 18, 2100–2111 (2002).
[CrossRef]

L. Lessard, M. West, D. MacMynowski, and S. Lall, “Warm-started wavefront reconstruction for adaptive optics,” J. Opt. Soc. Am. A 25, 1147–1155 (2008).
[CrossRef]

L. A. Poyneer and J.-P. Véran, “Optimal modal Fourier transform wave-front control,” J. Opt. Soc. Am. A 22, 1515–1526 (2005).
[CrossRef]

L. A. Poyneer, B. A. Macintosh, and J.-P. Véran, “Fourier-transform wavefront control with adaptive prediction of the atmosphere,” J. Opt. Soc. Am. A 24, 2645–2660 (2007).
[CrossRef]

B. L. Ellerbroek, “Efficient computation of minimum-variance wavefront reconstructors with sparse matrix techniques,” J. Opt. Soc. Am. A 19, 1803–1816 (2002).
[CrossRef]

L. Gilles, B. Ellerbroek, and C. Vogel, “multigrid preconditioned conjugate-gradient method for large-scale wave-front reconstruction,” J. Opt. Soc. Am. A 19, 1817–1822 (2002).
[CrossRef]

J.-P. Véran and G. Herriot, “Centroid gain compensation in Shack–Hartmann adaptive optics systems with natural or laser guide star,” J. Opt. Soc. Am. A 17, 1430–1439 (2000).
[CrossRef]

T. Nakajima, “Signal-to-noise ratio of the bispectral analysis of speckle interferometry,” J. Opt. Soc. Am. A 5, 1477–1491 (1988).
[CrossRef]

J.-F. Lavigne and J.-P. Véran, “woofer-tweeter control in an adaptive optics system using a Fourier reconstructor,” J. Opt. Soc. Am. A 25, 2271–2279 (2008).
[CrossRef]

Opt. Commun. (1)

D. L. Fried, “Adaptive optics wave function reconstruction and phase unwrapping when branch points are present,” Opt. Commun. 200, 43–72 (2001).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Proc. SPIE (3)

R. G. Dekany, M. C. Britton, D. T. Gavel, B. L. Ellerbroek, G. Herriot, C. E. Max, and J.-P. Véran, “Adaptive optics requirements for TMT,” Proc. SPIE 5490879–890 (2004).
[CrossRef]

S. A. Cornelissen, P. A. Bierden, and T. G. Bifano, “Development of a 4096 element deformable mirror for high contrast astronomical imaging,” Proc. SPIE 6306, 630606 (2006).
[CrossRef]

P. J. Hampton, R. Conan, C. Bradley, and P. Agathoklis, “Control of a woofer tweeter system of deformable mirrors,” Proc. SPIE 6274, 62741Z (2006).
[CrossRef]

Other (4)

J.-F. Lavigne, J.-P. Véran, and L. Poyneer, “Woofer–tweeter control algorithm for the Gemini planet imager,” in Adaptive Optics: Analysis and Methods; Computational Optical Sensing and Imaging; Digital Holography and Three-Dimensional Imaging; and Signal Recovery and Synthesis (Optical Society of America, 2007), paper AWB5.
[PubMed]

D. S. Watkins, Fundamentals of Matrix Computations (Wiley, 2002).
[CrossRef]

F. Roddier, Imaging through the Atmosphere (Cambridge University Press, 1999).

M. Vetterli and J. Kovacevic, Wavelets and Subband Coding (Prentice-Hall, 1995), Chap. 6.

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Figures (12)

Fig. 1
Fig. 1

Example interaction matrix for a pupil masked 16 × 16 DM and 15 × 15 SH-WFS.

Fig. 2
Fig. 2

Interaction matrix between 16 × 16 DM and 16 × 16 wavelet integration phase estimate. A similar interaction matrix is used for the wavelet-based system for the 32 × 32 DM case.

Fig. 3
Fig. 3

Approach to splitting the W-T signals using a wavelet-integration-based reconstruction by identifying a low-resolution phase error estimate for direct use as a woofer signal.

Fig. 4
Fig. 4

Interconnections between processes within one stage of wavelet integration showing input and output of each subsection.

Fig. 5
Fig. 5

W-T system that is SVD-based. Process is dominated by large non-sparse matrices.

Fig. 6
Fig. 6

Computational cost per tweeter actuator.

Fig. 7
Fig. 7

Actuation of a single actuator of the woofer (left) and tweeter (right). Diagram shows the difference in influence width of each mirror. Diagram is not to scale.

Fig. 8
Fig. 8

Step response with respect to Strehl ratio.

Fig. 9
Fig. 9

W-T stroke of 31 static phase screens.

Fig. 10
Fig. 10

W-T correction of 30 static phase screens.

Fig. 11
Fig. 11

Normalized rms error caused by CCD read-out noise.

Fig. 12
Fig. 12

Noiseless Strehl ratio when turbulence velocity is 25 m s . Sample rate is 1 kHz . The Strehl ratio is the ratio of peak light intensity of a corrected point source image to the peak of the image of an ideal point source with no turbulence.

Tables (1)

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Table 1 Measured Statistics of Strehl Ratio for Each Tested W-T Method a

Equations (39)

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[ φ x φ y ] = G φ + η ,
φ estimate = ( G T G ) 1 G T [ φ x φ y ] .
d = G φ atm + D a ,
d = G φ + D wfs , t a t + D wfs , w a w = G ( φ + D φ , t a t + D φ , w a w ) ,
D wfs , t = G D φ , t ,
D wfs , w = G D φ , w ,
D φ , t G D wfs , t ,
D φ , w G D wfs , w ,
2 Φ = ( Φ x u x + Φ y u y ) .
2 Φ estimate x 2 + 2 Φ estimate y 2 = ( Φ x + η x ) x + ( Φ y + η y ) y ,
[ 1 0 1 0 4 0 1 0 1 ] ( Φ estimate k ) = [ 1 1 1 1 ] ( X k ) + [ 1 1 1 1 ] ( Y k ) ,
Φ estimate k [ l + 1 ] = 1 4 ( [ 1 0 1 0 0 0 1 0 1 ] ( Φ estimate k [ l ] ) [ 1 1 1 1 ] ( X k ) [ 1 1 1 1 ] ( Y k ) ) .
H L ( z ) = ( 1 + z 1 ) 2 ,
H H ( z ) = ( 1 z 1 ) 2 .
Φ L H k 1 = 2 { Y k } ,
Φ H L k 1 = 2 { X k } ,
Φ H H k 2 = 2 4 { Y k H H ( z h 2 ) H L 2 ( z v ) } = 2 4 { X k H L 2 ( z h ) H H ( z v 2 ) } .
Y k 1 = 2 2 { Y k H L ( z h 2 ) H L 2 ( z v ) } ,
X k 1 = 2 2 { X k H L 2 ( z h ) H L ( z v 2 ) } .
Φ L L 0 = 0 ,
Φ H H 4 = null
Φ L L k + 1 = { H L ( z h ) ( H L ( z v ) 2 { Φ L L k } + H H ( z v ) 2 { Φ L H k } ) + H H ( z h ) ( H L ( z v ) 2 { Φ H L k } + H H ( z v ) 2 { Φ H H k } ) } .
Φ L L k + 1 = 1 2 { Φ L L k + 1 z h z v H ( z h 2 ) H ( z v 2 ) X k + 1 H H ( z h ) H L ( z v ) Y k + 1 H L ( z h ) H H ( z v ) } .
e t = φ err diag ( D φ , t ) D t , w e w ,
e w = D φ , w φ w .
a dm [ l ] = c dm a dm [ l 1 ] + g e dm [ l ] ,
D t , w = D wfs , t D wfs , w .
U t S t , w V w T = D t , w .
e = D wfs , t d ,
e w = V w S t , w U t T e ,
e t = ( I U t U t T ) e .
SNR = N pe N pe + N pix σ e 2 + N b ,
SNR N pe 8 σ e = 32 σ e .
Φ err = Φ atm + Φ cor ,
W = P tel exp ( j Φ err ) ,
[ d x , p , q d y , p , q ] = centroid ( | F { W p , q } | 2 + η p , q ) ,
d x = i , j x i , j I i , j i , j I i , j ,
d y = i , j y i , j I i , j i , j I i , j ,
[ φ ̇ x , p , q φ ̇ y , p , q ] α [ d x , p , q d y , p , q ] .

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