Abstract

The complexity of computing conventional matrix multiply wave-front reconstructors scales as O(n3) for most adaptive optical (AO) systems, where n is the number of deformable mirror (DM) actuators. This is impractical for proposed systems with extremely large n. It is known that sparse matrix methods improve this scaling for least-squares reconstructors, but sparse techniques are not immediately applicable to the minimum-variance reconstructors now favored for multiconjugate adaptive optical (MCAO) systems with multiple wave-front sensors (WFSs) and DMs. Complications arise from the nonsparse statistics of atmospheric turbulence, and the global tip/tilt WFS measurement errors associated with laser guide star (LGS) position uncertainty. A description is given of how sparse matrix methods can still be applied by use of a sparse approximation for turbulence statistics and by recognizing that the nonsparse matrix terms arising from LGS position uncertainty are low-rank adjustments that can be evaluated by using the matrix inversion lemma. Sample numerical results for AO and MCAO systems illustrate that the approximation made to turbulence statistics has negligible effect on estimation accuracy, the time to compute the sparse minimum-variance reconstructor for a conventional natural guide star AO system scales as O(n3/2) and is only a few seconds for n=3500, and sparse techniques reduce the reconstructor computations by a factor of 8 for sample MCAO systems with 2417 DM actuators and 4280 WFS subapertures. With extrapolation to 9700 actuators and 17,120 subapertures, a reduction by a factor of approximately 30 or 40 to 1 is predicted.

© 2002 Optical Society of America

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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]
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    [CrossRef]
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  24. The weighting matrix W derived in this section includes the global tip and tilt modes in the calculation of the mean square wave-front error. This is appropriate for AO applications to long-exposure imaging, where random, time-varying tip and tilt errors will degrade image quality in the same fashion as higher-order wave-front errors. For short-exposure imaging it may be more desirable to consider tip/tilt and higher-order wave-front aberrations separately. In this case the weighting matrix W for the mean square, higher-order wave-front error will still take the form defined by Eqs. (29)–(31), except that the matrix V1 now has three columns instead of one. The computational methods developed in the remainder of this paper are consequently still applicable.
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    [CrossRef]
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  27. Matlab 6 no longer supports the FLOPS command included in previous versions to report the exact number of floating-point operations required by an algorithm. No other computationally significant tasks were running on this dual-processor system during the timing tests reported in this paper.

2001

1994

1992

G. M. Cochran, “Sparse matrix techniques applied to deconvolution,” Comput. Elect. Eng. 18, 499–505 (1992).
[CrossRef]

1991

1990

G. Rousset, J.-C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Lena, C. Boyer, P. Jagourel, J.-P. Gaffard, F. Merkle, “First diffraction-limited astronomical images with adaptive optics,” Astron. Astrophys. 230, 29–32 (1990).

1983

1980

1977

1976

Agabi, A.

J. Vernin, A. Agabi, R. Avila, M. Azouit, R. Conan, F. Martin, E. Masciadri, L. Sanchez, A. Ziad, “1998 Gemini site testing campaign: Cerro Pachon and Cerro Tololo,” (Gemini Observatory, Hilo, Hawaii, 2000).

Avila, R.

J. Vernin, A. Agabi, R. Avila, M. Azouit, R. Conan, F. Martin, E. Masciadri, L. Sanchez, A. Ziad, “1998 Gemini site testing campaign: Cerro Pachon and Cerro Tololo,” (Gemini Observatory, Hilo, Hawaii, 2000).

Azouit, M.

J. Vernin, A. Agabi, R. Avila, M. Azouit, R. Conan, F. Martin, E. Masciadri, L. Sanchez, A. Ziad, “1998 Gemini site testing campaign: Cerro Pachon and Cerro Tololo,” (Gemini Observatory, Hilo, Hawaii, 2000).

Beckers, J. M.

J. M. Beckers, “Increasing the size of the isoplanatic patch with multi-conjugate adaptive optics,” in Proceedings of European Southern Observatory Conference and Workshop on Very Large Telescopes and Their Instrumentation, M.-H. Ulrich, ed., Vol. 30 of ESO Conference and Workshop Proceedings (European Southern Observatory, Garching, Germany, 1988), pp. 693–703.

Boyer, C.

G. Rousset, J.-C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Lena, C. Boyer, P. Jagourel, J.-P. Gaffard, F. Merkle, “First diffraction-limited astronomical images with adaptive optics,” Astron. Astrophys. 230, 29–32 (1990).

Cochran, G. M.

G. M. Cochran, “Sparse matrix techniques applied to deconvolution,” Comput. Elect. Eng. 18, 499–505 (1992).
[CrossRef]

G. M. Cochran, “Sparse matrix techniques in wavefront reconstruction,” (Optical Sciences Company, Anaheim, Calif., 1986).

Conan, J.-M.

Conan, R.

J. Vernin, A. Agabi, R. Avila, M. Azouit, R. Conan, F. Martin, E. Masciadri, L. Sanchez, A. Ziad, “1998 Gemini site testing campaign: Cerro Pachon and Cerro Tololo,” (Gemini Observatory, Hilo, Hawaii, 2000).

Ellerbroek, B. L.

B. L. Ellerbroek, “Methods for correcting tilt anisoplanatism in laser-guide-star-based multi-conjugate adaptive optics,” J. Opt. Soc. Am. A 18, 2539–2547 (2001).
[CrossRef]

B. L. Ellerbroek, “First order performance evaluation of adaptive-optics systems for atmospheric turbulence compensation in extended field-of-view astronomical telescopes,” J. Opt. Soc. Am. A 11, 783–805 (1994).
[CrossRef]

L. Gilles, C. R. Vogel, B. L. Ellerbroek, “Iterative algorithms for large scale wave front reconstruction,” in Signal Recovery and Synthesis, Vol. 67 of OSA Trends in Optics and Photonics (Optical Society of America, Washington D.C.,2001), pp. 100–101.

B. L. Ellerbroek, “Comparison of least squares and minimum variance wavefront reconstruction for atmospheric turbulence compensation in the presence of noise,” (Optical Sciences Company, Anaheim, Calif., 1986).

R. Flicker, F. J. Rigaut, B. L. Ellerbroek, “Comparison of multi-conjugate adaptive optics configurations and control algorithms for the Gemini South 8-m telescope,” in Adaptive Optical Systems Technology, P. L. Wizinowich, ed., Proc. SPIE4007, 1032–1043 (2000).
[CrossRef]

Flicker, R.

R. Flicker, F. J. Rigaut, B. L. Ellerbroek, “Comparison of multi-conjugate adaptive optics configurations and control algorithms for the Gemini South 8-m telescope,” in Adaptive Optical Systems Technology, P. L. Wizinowich, ed., Proc. SPIE4007, 1032–1043 (2000).
[CrossRef]

Fontanella, J.-C.

G. Rousset, J.-C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Lena, C. Boyer, P. Jagourel, J.-P. Gaffard, F. Merkle, “First diffraction-limited astronomical images with adaptive optics,” Astron. Astrophys. 230, 29–32 (1990).

Fried, D. L.

Fusco, T.

Gaffard, J.-P.

G. Rousset, J.-C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Lena, C. Boyer, P. Jagourel, J.-P. Gaffard, F. Merkle, “First diffraction-limited astronomical images with adaptive optics,” Astron. Astrophys. 230, 29–32 (1990).

Gardner, C. S.

George, A.

A. George, J. Liu, Computer Solutions of Large Symmetric Positive Definite Systems (Prentice-Hall, Englewood Cliffs, N.J., 1981).

Gigan, P.

G. Rousset, J.-C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Lena, C. Boyer, P. Jagourel, J.-P. Gaffard, F. Merkle, “First diffraction-limited astronomical images with adaptive optics,” Astron. Astrophys. 230, 29–32 (1990).

Gilles, L.

L. Gilles, C. R. Vogel, B. L. Ellerbroek, “Iterative algorithms for large scale wave front reconstruction,” in Signal Recovery and Synthesis, Vol. 67 of OSA Trends in Optics and Photonics (Optical Society of America, Washington D.C.,2001), pp. 100–101.

Golub, G. H.

G. H. Golub, C. F. van Loan, Matrix Computations (John Hopkins U. Press, Baltimore, Md., 1996), p. 18.

Hardy, J. W.

Hermann, J.

Hudgin, R.

Jagourel, P.

G. Rousset, J.-C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Lena, C. Boyer, P. Jagourel, J.-P. Gaffard, F. Merkle, “First diffraction-limited astronomical images with adaptive optics,” Astron. Astrophys. 230, 29–32 (1990).

Johnston, D. C.

Kern, P.

G. Rousset, J.-C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Lena, C. Boyer, P. Jagourel, J.-P. Gaffard, F. Merkle, “First diffraction-limited astronomical images with adaptive optics,” Astron. Astrophys. 230, 29–32 (1990).

Koliopoulous, C. L.

Lefevbre, J. E.

Lena, P.

G. Rousset, J.-C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Lena, C. Boyer, P. Jagourel, J.-P. Gaffard, F. Merkle, “First diffraction-limited astronomical images with adaptive optics,” Astron. Astrophys. 230, 29–32 (1990).

Levine, B. M.

L. Schmutz, B. M. Levine, A. Wirth, C. Strandley, “Adaptive optics concepts for extremely large aperture telescopes,” in Bäckaskog Workshop on Extremely Large Telescopes, T. Andersen, A. Ardeberg, R. Gilmozzi, eds., Vol. 57 of ESO Conference and Workshop Proceedings (European Southern Observatory, Garching, Germany, 1999), pp. 217–223.

Liu, J.

A. George, J. Liu, Computer Solutions of Large Symmetric Positive Definite Systems (Prentice-Hall, Englewood Cliffs, N.J., 1981).

Martin, F.

J. Vernin, A. Agabi, R. Avila, M. Azouit, R. Conan, F. Martin, E. Masciadri, L. Sanchez, A. Ziad, “1998 Gemini site testing campaign: Cerro Pachon and Cerro Tololo,” (Gemini Observatory, Hilo, Hawaii, 2000).

Masciadri, E.

J. Vernin, A. Agabi, R. Avila, M. Azouit, R. Conan, F. Martin, E. Masciadri, L. Sanchez, A. Ziad, “1998 Gemini site testing campaign: Cerro Pachon and Cerro Tololo,” (Gemini Observatory, Hilo, Hawaii, 2000).

Merkle, F.

G. Rousset, J.-C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Lena, C. Boyer, P. Jagourel, J.-P. Gaffard, F. Merkle, “First diffraction-limited astronomical images with adaptive optics,” Astron. Astrophys. 230, 29–32 (1990).

Michau, V.

Mugnier, L. M.

Noll, R. J.

Pissantesky, S.

S. Pissantesky, Sparse Matrix Technology (Academic, Orlando, Fla., 1984), Chap. 4.

Rigaut, F.

G. Rousset, J.-C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Lena, C. Boyer, P. Jagourel, J.-P. Gaffard, F. Merkle, “First diffraction-limited astronomical images with adaptive optics,” Astron. Astrophys. 230, 29–32 (1990).

Rigaut, F. J.

R. Flicker, F. J. Rigaut, B. L. Ellerbroek, “Comparison of multi-conjugate adaptive optics configurations and control algorithms for the Gemini South 8-m telescope,” in Adaptive Optical Systems Technology, P. L. Wizinowich, ed., Proc. SPIE4007, 1032–1043 (2000).
[CrossRef]

Roggemann, M. C.

M. C. Roggemann, B. Welsh, Imaging through Turbulence (CRC Press, Boca Raton, Fla., 1996).

Rousset, G.

T. Fusco, J.-M. Conan, G. Rousset, L. M. Mugnier, V. Michau, “Optimal wave-front reconstruction strategies for multi-conjugate adaptive optics,” J. Opt. Soc. Am. A 18, 2527–2538 (2001).
[CrossRef]

G. Rousset, J.-C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Lena, C. Boyer, P. Jagourel, J.-P. Gaffard, F. Merkle, “First diffraction-limited astronomical images with adaptive optics,” Astron. Astrophys. 230, 29–32 (1990).

Sanchez, L.

J. Vernin, A. Agabi, R. Avila, M. Azouit, R. Conan, F. Martin, E. Masciadri, L. Sanchez, A. Ziad, “1998 Gemini site testing campaign: Cerro Pachon and Cerro Tololo,” (Gemini Observatory, Hilo, Hawaii, 2000).

Schmutz, L.

L. Schmutz, B. M. Levine, A. Wirth, C. Strandley, “Adaptive optics concepts for extremely large aperture telescopes,” in Bäckaskog Workshop on Extremely Large Telescopes, T. Andersen, A. Ardeberg, R. Gilmozzi, eds., Vol. 57 of ESO Conference and Workshop Proceedings (European Southern Observatory, Garching, Germany, 1999), pp. 217–223.

Strandley, C.

L. Schmutz, B. M. Levine, A. Wirth, C. Strandley, “Adaptive optics concepts for extremely large aperture telescopes,” in Bäckaskog Workshop on Extremely Large Telescopes, T. Andersen, A. Ardeberg, R. Gilmozzi, eds., Vol. 57 of ESO Conference and Workshop Proceedings (European Southern Observatory, Garching, Germany, 1999), pp. 217–223.

Tyler, G. A.

van Loan, C. F.

G. H. Golub, C. F. van Loan, Matrix Computations (John Hopkins U. Press, Baltimore, Md., 1996), p. 18.

Vernin, J.

J. Vernin, A. Agabi, R. Avila, M. Azouit, R. Conan, F. Martin, E. Masciadri, L. Sanchez, A. Ziad, “1998 Gemini site testing campaign: Cerro Pachon and Cerro Tololo,” (Gemini Observatory, Hilo, Hawaii, 2000).

Vogel, C. R.

L. Gilles, C. R. Vogel, B. L. Ellerbroek, “Iterative algorithms for large scale wave front reconstruction,” in Signal Recovery and Synthesis, Vol. 67 of OSA Trends in Optics and Photonics (Optical Society of America, Washington D.C.,2001), pp. 100–101.

Wallner, E. P.

Walsh, B. M.

Welsh, B.

M. C. Roggemann, B. Welsh, Imaging through Turbulence (CRC Press, Boca Raton, Fla., 1996).

Welsh, B. M.

Wirth, A.

L. Schmutz, B. M. Levine, A. Wirth, C. Strandley, “Adaptive optics concepts for extremely large aperture telescopes,” in Bäckaskog Workshop on Extremely Large Telescopes, T. Andersen, A. Ardeberg, R. Gilmozzi, eds., Vol. 57 of ESO Conference and Workshop Proceedings (European Southern Observatory, Garching, Germany, 1999), pp. 217–223.

Ziad, A.

J. Vernin, A. Agabi, R. Avila, M. Azouit, R. Conan, F. Martin, E. Masciadri, L. Sanchez, A. Ziad, “1998 Gemini site testing campaign: Cerro Pachon and Cerro Tololo,” (Gemini Observatory, Hilo, Hawaii, 2000).

Astron. Astrophys.

G. Rousset, J.-C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Lena, C. Boyer, P. Jagourel, J.-P. Gaffard, F. Merkle, “First diffraction-limited astronomical images with adaptive optics,” Astron. Astrophys. 230, 29–32 (1990).

Comput. Elect. Eng.

G. M. Cochran, “Sparse matrix techniques applied to deconvolution,” Comput. Elect. Eng. 18, 499–505 (1992).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Other

G. M. Cochran, “Sparse matrix techniques in wavefront reconstruction,” (Optical Sciences Company, Anaheim, Calif., 1986).

L. Schmutz, B. M. Levine, A. Wirth, C. Strandley, “Adaptive optics concepts for extremely large aperture telescopes,” in Bäckaskog Workshop on Extremely Large Telescopes, T. Andersen, A. Ardeberg, R. Gilmozzi, eds., Vol. 57 of ESO Conference and Workshop Proceedings (European Southern Observatory, Garching, Germany, 1999), pp. 217–223.

L. Gilles, C. R. Vogel, B. L. Ellerbroek, “Iterative algorithms for large scale wave front reconstruction,” in Signal Recovery and Synthesis, Vol. 67 of OSA Trends in Optics and Photonics (Optical Society of America, Washington D.C.,2001), pp. 100–101.

B. L. Ellerbroek, “Comparison of least squares and minimum variance wavefront reconstruction for atmospheric turbulence compensation in the presence of noise,” (Optical Sciences Company, Anaheim, Calif., 1986).

J. M. Beckers, “Increasing the size of the isoplanatic patch with multi-conjugate adaptive optics,” in Proceedings of European Southern Observatory Conference and Workshop on Very Large Telescopes and Their Instrumentation, M.-H. Ulrich, ed., Vol. 30 of ESO Conference and Workshop Proceedings (European Southern Observatory, Garching, Germany, 1988), pp. 693–703.

R. Flicker, F. J. Rigaut, B. L. Ellerbroek, “Comparison of multi-conjugate adaptive optics configurations and control algorithms for the Gemini South 8-m telescope,” in Adaptive Optical Systems Technology, P. L. Wizinowich, ed., Proc. SPIE4007, 1032–1043 (2000).
[CrossRef]

G. H. Golub, C. F. van Loan, Matrix Computations (John Hopkins U. Press, Baltimore, Md., 1996), p. 18.

A. George, J. Liu, Computer Solutions of Large Symmetric Positive Definite Systems (Prentice-Hall, Englewood Cliffs, N.J., 1981).

S. Pissantesky, Sparse Matrix Technology (Academic, Orlando, Fla., 1984), Chap. 4.

The weighting matrix W derived in this section includes the global tip and tilt modes in the calculation of the mean square wave-front error. This is appropriate for AO applications to long-exposure imaging, where random, time-varying tip and tilt errors will degrade image quality in the same fashion as higher-order wave-front errors. For short-exposure imaging it may be more desirable to consider tip/tilt and higher-order wave-front aberrations separately. In this case the weighting matrix W for the mean square, higher-order wave-front error will still take the form defined by Eqs. (29)–(31), except that the matrix V1 now has three columns instead of one. The computational methods developed in the remainder of this paper are consequently still applicable.

J. Vernin, A. Agabi, R. Avila, M. Azouit, R. Conan, F. Martin, E. Masciadri, L. Sanchez, A. Ziad, “1998 Gemini site testing campaign: Cerro Pachon and Cerro Tololo,” (Gemini Observatory, Hilo, Hawaii, 2000).

Matlab 6 no longer supports the FLOPS command included in previous versions to report the exact number of floating-point operations required by an algorithm. No other computationally significant tasks were running on this dual-processor system during the timing tests reported in this paper.

M. C. Roggemann, B. Welsh, Imaging through Turbulence (CRC Press, Boca Raton, Fla., 1996).

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Figures (3)

Fig. 1
Fig. 1

Computation requirements for conventional and sparse calculations of classical least-squares reconstruction algorithms. These results are for a square-aperture geometry and the so-called “Hudgins” or “shearing interferometer” wave-front sensor geometry. The number of floating-point operations (FLOPs) needed to compute the control algorithm by using a conventional matrix inversion has been evaluated for systems of order 100, 225, and 400 and extrapolated by using the predicted third-order power law. The number of operations necessary for the sparse matrix factorization has been explicitly computed for AO systems of up to order 90,000 and scales with the three-halves power of the order of the system.

Fig. 2
Fig. 2

Influence matrix models. Part (a) illustrates the relationship between the turbulence phase screen vector x, the DM actuator command vector a, and the residual phase error vector ϕ. These three vectors are defined as values on grids of points in the planes of the phase screens, the DM conjugate locations, and the telescope aperture, respectively. The influence matrices Hx and Ha are defined by tracing rays through the phase screens and mirrors as illustrated. Part (b) illustrates the similar relationship between the phase screen vector x and the WFS measurement vector s. In this case rays are traced from the guide star(s) through the phase screen(s) to obtain a wave front in the telescope aperture plane, and the WFS measurements are then computed as the average x and y wave-front gradients over each subaperture.

Fig. 3
Fig. 3

Discrete Laplacian operator. This figure illustrates the coefficients for two rows of the discrete Laplacian, or curvature, operator C appearing in relation (51). The dots represent the grid points of the discrete turbulence layer. The values printed in a regular font are the five nonzero coefficients needed to compute the curvature of the phase profile at the interior grid point A. The italicized values are the nonzero coefficients needed to compute the curvature at the grid point B on the boundary of the phase profile. The coefficients that should be assigned to grid points lying outside the boundary have been “folded over” back into the grid so that the sum of the coefficients remains zero.

Tables (5)

Tables Icon

Table 1 Atmospheric Turbulence Profile Used for Simulationsa

Tables Icon

Table 2 Simulated AO System Parametersa

Tables Icon

Table 3 Reconstructor Performance versus WFS Measurement Noise for an Order 8×8 NGS AO Systema

Tables Icon

Table 4 SRA Wave-Front Fitting Error versus Order of Correction for a Conventional NGS AO Systema

Tables Icon

Table 5 Results and Scaling Law Predictions for CMMR and SRA Performance for MCAO Systemsa

Equations (67)

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

s=Gaa+n.
aˆ=arg minas-Gaa2,
aˆ=(GaTGa)GaTs,
GaTGa=LLT,
v=GaTs,Lw=v,LTaˆ=w.
ϕ=Hxx-Haa.
σ2=ϕTWϕ,
a=Es,
s=Gxx+n,
E*=arg minEσ2+ka2,
σ2+ka2=(Hxx-HaEs)TW(Hxx-HaEs)+ksTETEs
0=2σ2+ka2EijE=E*.
s(HaTWHxx)T=s(HaTWHaE*s)T+ks(E*s)T.
Cvw=vwT
(HaTWHx)Cxs=(HaTWHa+kI)E*Css.
NwTE*=0,
(HaTWHx)Cxs=(HaTWHa+NwNwT+kI)E*Css.
E*=(HaTWHa+NwNwT+kI)-1(HaTWHx)CxsCss-1=FxEx,
Fx=(HaTWHa+NwNwT+kI)-1(HaTWHx),
Ex=CxsCss-1.
ϕ=ϕ1ϕn,
φi(r)=jϕjiej(r),
ϕTWϕ=i=1nwi(ϕi)TVϕi,
(ϕi)TVϕi= drA(r)φi(r)- drA(r)φi(r)2,
 drA(r)=1.
V=V0-V1V1T,
(V0)ij= drA(r)ei(r)ej(r),
(V1)i= drA(r)ei(r).
W=W0-W1W1T,
W0=diag(w1V0, , wnV0),
W1=diag(w1V1, , wnV1).
u=HaTW Hxv,
HaTW Hx=M-UVT,
M=HaTW0Hx,
U=HaTW1,
V=HxTW1.
u=Mv-U(VTv),
u=(HaTWHa+NwNwT+kI)-1u
HaTWHa+NwNwT+kI=M-UVT,
M=HaTW0Ha+kI,
U=(HaTW1Nw),
V=(HaTW1-Nw).
(MUVT)-1=M-1±M-1U(IVTM-1U)-1(M-1V)T,
u=M-1u+((M-1U){(I-VTM-1U)-1[(M-1V)Tu]}).
Ex=CxsCss-1=CxxGxT(GxCxxTGxT+Cnn)-1=(GxTCnn-1Gx+Cxx-1)-1GxTCnn-1,
Lij=1ifphasepointiislocatedonscreenj0otherwise,
Zij=1ifphasepointiistheoriginofscreenj0otherwise.
(GxTCnn-1Gx+Cxx-1+ZZT)(I-LZT)=GxTCnn-1Gx+Cxx-1-Cxx-1LZTGxTCnn-1Gx+Cxx-1,
(GxTCnn-1Gx+Cxx-1+ZZT)-1(I-LZT)(GxTCnn-1Gx+Cxx-1)-1.
uTCxx-1v=uTxxT-1v= drdru(r)v(r)[x(r)xT(r)-1]= dκdκuˆ(κ)vˆ*(κ)[xˆ(κ)xˆ*(κ)-1] dκuˆ(κ)vˆ*(κ)κ11/3 dκ[κ2uˆ(κ)][κ2vˆ(κ)]* dr2u(r)2v(r),
Cxx-1CTC,
Ex=(GxTCnn-1Gx+CTC+ZZT)-1GxTCnn-1.
s=shst=GhGtx+nhnt,
Cnn=Nh+σt2TTT00Nt.
(Nh+σt2TTT)-1=Nh-1-σt2Nh-1T(I+σt2TTNh-1T)-1(Nh-1T)TNh-1-Nh-1T(TTNh-1T)-1(Nh-1T)Tasσt2.
u=GxTCnn-1v.
GxTCnn-1=[GhTNh-1(I-TPT)GtTNt-1],
PT=(TTNh-1T)-1TTNh-1.
GxTCnn-1=M-UVT,
M=[GhTNh-1GtTNt-1],
U=GhTNh-1T,
V=PTT0=Nh-1T(TTNh-1T)-10.
u=(GxTCnn-1Gx+CTC+ZZT)-1u.
GxTCnn-1Gx+CTC+ZZT=M-UVT,
M=GhTNh-1Gh+CTC+ZZT,
U=[GhTNh-1T-GtTNt-1],
V=[GhTPTTGtT].

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