Abstract

Wavefront sensorless adaptive optics methodologies are widely considered in scanning fluorescence microscopy where direct wavefront sensing is challenging. In these methodologies, aberration correction is performed by sequentially changing the settings of the adaptive element until a predetermined image quality metric is optimized. An efficient aberration correction can be achieved by modeling the image quality metric with a quadratic polynomial. We propose a new method to compute the parameters of the polynomial from experimental data. This method guarantees that the quadratic form in the polynomial is semidefinite, resulting in a more robust computation of the parameters with respect to existing methods. In addition, we propose an algorithm to perform aberration correction requiring a minimum of N+1 measurements, where N is the number of considered aberration modes. This algorithm is based on a closed-form expression for the exact optimization of the quadratic polynomial. Our arguments are corroborated by experimental validation in a laboratory environment.

© 2012 Optical Society of America

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2012

S. A. Rahman and M. J. Booth, “Adaptive optics for high-resolution microscopy: wave front sensing using back scattered light,” Proc. SPIE 8253, 82530I (2012).
[CrossRef]

T. van Werkhoven, H. Truong, J. Antonello, R. Fraanje, H. Gerritsen, M. Verhaegen, and C. Keller, “Coherence-gated wavefront sensing for microscopy using fringe analysis,” Proc. SPIE 8253, 82530E (2012).
[CrossRef]

D. Débarre, A. Facomprez, and E. Beaurepaire, “Assessing correction accuracy in image-based adaptive optics,” Proc. SPIE 8253, 82530F (2012).
[CrossRef]

J. Antonello, R. Fraanje, H. Song, M. Verhaegen, H. Gerritsen, C. U. Keller, and T. van Werkhoven, “Data driven identification and aberration correction for model based sensorless adaptive optics,” Proc. SPIE 8436, 84360S (2012).
[CrossRef]

A. Facomprez, E. Beaurepaire, and D. Débarre, “Accuracy of correction in modal sensorless adaptive optics,” Opt. Express 20, 2598–2612 (2012).
[CrossRef]

2011

2010

2009

2008

2007

2006

2005

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

A. J. Wright, D. Burns, B. A. Patterson, S. P. Poland, G. J. Valentine, and J. M. Girkin, “Exploration of the optimisation algorithms used in the implementation of adaptive optics in confocal and multiphoton microscopy,” Microsc. Res. Tech. 67, 36–44 (2005).
[CrossRef]

2004

M. J. Booth, “Wavefront sensorless adaptive optics, modal wavefront sensing, and sphere packings,” Proc. SPIE 5553, 150–158 (2004).
[CrossRef]

M. Feierabend, M. Rückel, and W. Denk, “Coherence-gated wave-front sensing in strongly scattering samples,” Opt. Lett. 29, 2255–2257 (2004).
[CrossRef]

2003

2002

2000

1999

J. F. Sturm, “Using SEDUMI 1.02, a MATLAB toolbox for optimization over symmetric cones,” Optim. Methods Softw. 11, 625–653 (1999).
[CrossRef]

1998

G. Vdovin, “Optimization-based operation of micromachined deformable mirrors,” Proc. SPIE 3353, 902–909 (1998).
[CrossRef]

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder–Mead simplex method in low dimensions,” SIAM J. Optim. 9, 112–147 (1998).
[CrossRef]

1996

L. Vandenberghe and S. Boyd, “Semidefinite programming,” SIAM Rev. 38, 49–95 (1996).
[CrossRef]

1994

G.-M. Dai, “Modified Hartmann-Shack wavefront sensing and iterative wavefront reconstruction,” Proc. SPIE 2201, 562–573 (1994).
[CrossRef]

1990

W. Denk, J. Strickler, and W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248, 73–76 (1990).
[CrossRef]

1976

Albert, O.

L. Sherman, J. Y. Ye, O. Albert, and T. B. Norris, “Adaptive correction of depth-induced aberrations in multiphoton scanning microscopy using a deformable mirror,” J. Microsc. 206, 65–71 (2002).
[CrossRef]

O. Albert, L. Sherman, G. Mourou, T. B. Norris, and G. Vdovin, “Smart microscope: an adaptive optics learning system for aberration correction in multiphoton confocal microscopy,” Opt. Lett. 25, 52–54 (2000).
[CrossRef]

Andilla, J.

Antonello, J.

J. Antonello, R. Fraanje, H. Song, M. Verhaegen, H. Gerritsen, C. U. Keller, and T. van Werkhoven, “Data driven identification and aberration correction for model based sensorless adaptive optics,” Proc. SPIE 8436, 84360S (2012).
[CrossRef]

T. van Werkhoven, H. Truong, J. Antonello, R. Fraanje, H. Gerritsen, M. Verhaegen, and C. Keller, “Coherence-gated wavefront sensing for microscopy using fringe analysis,” Proc. SPIE 8253, 82530E (2012).
[CrossRef]

Artigas, D.

Aviles-Espinosa, R.

Azucena, O.

Ballesta, J.

J. W. Cha, J. Ballesta, and P. T. C. So, “Shack–Hartmann wavefront-sensor-based adaptive optics system for multiphoton microscopy,” J. Biomed. Opt. 15, 046022 (2010).
[CrossRef]

Beaurepaire, E.

Betzig, E.

N. Ji, D. E. Milkie, and E. Betzig, “Adaptive optics via pupil segmentation for high-resolution imaging in biological tissues,” Nat. Methods 7, 141–147 (2009).
[CrossRef]

Booth, M. J.

S. A. Rahman and M. J. Booth, “Adaptive optics for high-resolution microscopy: wave front sensing using back scattered light,” Proc. SPIE 8253, 82530I (2012).
[CrossRef]

B. Wang, and M. J. Booth, “Optimum deformable mirror modes for sensorless adaptive optics,” Opt. Commun. 282, 4467–4474 (2009).
[CrossRef]

D. Débarre, E. J. Botcherby, T. Watanabe, S. Srinivas, M. J. Booth, and T. Wilson, “Image-based adaptive optics for two-photon microscopy,” Opt. Lett. 34, 2495–2497 (2009).
[CrossRef]

D. Débarre, E. J. Botcherby, M. J. Booth, and T. Wilson, “Adaptive optics for structured illumination microscopy,” Opt. Express 16, 9290–9305 (2008).
[CrossRef]

M. J. Booth, “Adaptive optics in microscopy,” Philos. Trans. Ser. A 365, 2829–2843 (2007).
[CrossRef]

D. Débarre, M. J. Booth, and T. Wilson, “Image based adaptive optics through optimisation of low spatial frequencies,” Opt. Express 15, 8176–8190 (2007).
[CrossRef]

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

M. J. Booth, “Wave front sensor-less adaptive optics: a model-based approach using sphere packings,” Opt. Express 14, 1339–1352 (2006).
[CrossRef]

M. J. Booth, “Wavefront sensorless adaptive optics, modal wavefront sensing, and sphere packings,” Proc. SPIE 5553, 150–158 (2004).
[CrossRef]

M. A. A. Neil, M. J. Booth, and T. Wilson, “New modal wave-front sensor: a theoretical analysis,” J. Opt. Soc. Am. A 17, 1098–1107 (2000).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

Botcherby, E. J.

Boyd, S.

L. Vandenberghe and S. Boyd, “Semidefinite programming,” SIAM Rev. 38, 49–95 (1996).
[CrossRef]

Burns, D.

W. Lubeigt, S. P. Poland, G. J. Valentine, A. J. Wright, J. M. Girkin, and D. Burns, “Search-based active optic systems for aberration correction in time-independent applications,” Appl. Opt. 49, 307–314 (2010).
[CrossRef]

A. J. Wright, D. Burns, B. A. Patterson, S. P. Poland, G. J. Valentine, and J. M. Girkin, “Exploration of the optimisation algorithms used in the implementation of adaptive optics in confocal and multiphoton microscopy,” Microsc. Res. Tech. 67, 36–44 (2005).
[CrossRef]

P. Marsh, D. Burns, and J. Girkin, “Practical implementation of adaptive optics in multiphoton microscopy,” Opt. Express 11, 1123–1130 (2003).
[CrossRef]

Cao, J.

Cha, J. W.

J. W. Cha, J. Ballesta, and P. T. C. So, “Shack–Hartmann wavefront-sensor-based adaptive optics system for multiphoton microscopy,” J. Biomed. Opt. 15, 046022 (2010).
[CrossRef]

Chen, D. C.

Crest, J.

Dai, G.-M.

G.-M. Dai, “Modified Hartmann-Shack wavefront sensing and iterative wavefront reconstruction,” Proc. SPIE 2201, 562–573 (1994).
[CrossRef]

Dainty, J. C.

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

Daly, E.

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

Débarre, D.

Denk, W.

M. Feierabend, M. Rückel, and W. Denk, “Coherence-gated wave-front sensing in strongly scattering samples,” Opt. Lett. 29, 2255–2257 (2004).
[CrossRef]

W. Denk, J. Strickler, and W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248, 73–76 (1990).
[CrossRef]

Dillon, D.

Evans, C. L.

Facomprez, A.

A. Facomprez, E. Beaurepaire, and D. Débarre, “Accuracy of correction in modal sensorless adaptive optics,” Opt. Express 20, 2598–2612 (2012).
[CrossRef]

D. Débarre, A. Facomprez, and E. Beaurepaire, “Assessing correction accuracy in image-based adaptive optics,” Proc. SPIE 8253, 82530F (2012).
[CrossRef]

Feierabend, M.

Fernandez, B.

Fienup, J. R.

Fraanje, R.

J. Antonello, R. Fraanje, H. Song, M. Verhaegen, H. Gerritsen, C. U. Keller, and T. van Werkhoven, “Data driven identification and aberration correction for model based sensorless adaptive optics,” Proc. SPIE 8436, 84360S (2012).
[CrossRef]

T. van Werkhoven, H. Truong, J. Antonello, R. Fraanje, H. Gerritsen, M. Verhaegen, and C. Keller, “Coherence-gated wavefront sensing for microscopy using fringe analysis,” Proc. SPIE 8253, 82530E (2012).
[CrossRef]

H. Song, R. Fraanje, G. Schitter, H. Kroese, G. Vdovin, and M. Verhaegen, “Model-based aberration correction in a closed-loop wavefront-sensor-less adaptive optics system,” Opt. Express 18, 24070–24084 (2010).
[CrossRef]

Freudiger, C. W.

Fu, M.

Garcia, D.

Gavel, D.

Gerritsen, H.

T. van Werkhoven, H. Truong, J. Antonello, R. Fraanje, H. Gerritsen, M. Verhaegen, and C. Keller, “Coherence-gated wavefront sensing for microscopy using fringe analysis,” Proc. SPIE 8253, 82530E (2012).
[CrossRef]

J. Antonello, R. Fraanje, H. Song, M. Verhaegen, H. Gerritsen, C. U. Keller, and T. van Werkhoven, “Data driven identification and aberration correction for model based sensorless adaptive optics,” Proc. SPIE 8436, 84360S (2012).
[CrossRef]

Girkin, J.

Girkin, J. M.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed.(Roberts & Company, 2004).

Hardy, J. W.

J. W. Hardy, Adaptive Optics for Astronomical Telescopes(Oxford University, 1998).

Henrion, D.

Y. Labit, D. Peaucelle, and D. Henrion, “SEDUMI INTERFACE 1.02: a tool for solving LMI problems with SEDUMI,” in Proceedings of the 2002 IEEE International Symposium on Computer Aided Control System Design (IEEE, 2002), pp. 272–277.

Ji, N.

N. Ji, D. E. Milkie, and E. Betzig, “Adaptive optics via pupil segmentation for high-resolution imaging in biological tissues,” Nat. Methods 7, 141–147 (2009).
[CrossRef]

Keller, C.

T. van Werkhoven, H. Truong, J. Antonello, R. Fraanje, H. Gerritsen, M. Verhaegen, and C. Keller, “Coherence-gated wavefront sensing for microscopy using fringe analysis,” Proc. SPIE 8253, 82530E (2012).
[CrossRef]

Keller, C. U.

J. Antonello, R. Fraanje, H. Song, M. Verhaegen, H. Gerritsen, C. U. Keller, and T. van Werkhoven, “Data driven identification and aberration correction for model based sensorless adaptive optics,” Proc. SPIE 8436, 84360S (2012).
[CrossRef]

Kner, P.

Kroese, H.

Kubby, J.

Labit, Y.

Y. Labit, D. Peaucelle, and D. Henrion, “SEDUMI INTERFACE 1.02: a tool for solving LMI problems with SEDUMI,” in Proceedings of the 2002 IEEE International Symposium on Computer Aided Control System Design (IEEE, 2002), pp. 272–277.

Lagarias, J. C.

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder–Mead simplex method in low dimensions,” SIAM J. Optim. 9, 112–147 (1998).
[CrossRef]

Leray, A.

Levecq, X.

Linhai, H.

Löfberg, J.

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

Loktev, M.

G. Vdovin and M. Loktev, “Deformable mirror with thermal actuators,” Opt. Lett. 27, 677–679 (2002).
[CrossRef]

M. Loktev, O. Soloviev, and G. Vdovin, Adaptive Optics Guide, 3rd ed. (OKO Technologies, 2008).

Loza-Alvarez, P.

Lubeigt, W.

Marsh, P.

Mertz, J.

Milkie, D. E.

N. Ji, D. E. Milkie, and E. Betzig, “Adaptive optics via pupil segmentation for high-resolution imaging in biological tissues,” Nat. Methods 7, 141–147 (2009).
[CrossRef]

Miller, J. J.

Mourou, G.

Murray, L.

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

L. Murray, “Smart optics: wavefront sensor-less adaptive optics—image correction through sharpness maximisation,” Ph.D. thesis (National University of Ireland, 2007).

Neil, M. A. A.

Nieto, M.

Noll, R. J.

Norris, T. B.

L. Sherman, J. Y. Ye, O. Albert, and T. B. Norris, “Adaptive correction of depth-induced aberrations in multiphoton scanning microscopy using a deformable mirror,” J. Microsc. 206, 65–71 (2002).
[CrossRef]

O. Albert, L. Sherman, G. Mourou, T. B. Norris, and G. Vdovin, “Smart microscope: an adaptive optics learning system for aberration correction in multiphoton confocal microscopy,” Opt. Lett. 25, 52–54 (2000).
[CrossRef]

Olarte, O. E.

Olivier, N.

Olivier, S.

Patterson, B. A.

A. J. Wright, D. Burns, B. A. Patterson, S. P. Poland, G. J. Valentine, and J. M. Girkin, “Exploration of the optimisation algorithms used in the implementation of adaptive optics in confocal and multiphoton microscopy,” Microsc. Res. Tech. 67, 36–44 (2005).
[CrossRef]

Peaucelle, D.

Y. Labit, D. Peaucelle, and D. Henrion, “SEDUMI INTERFACE 1.02: a tool for solving LMI problems with SEDUMI,” in Proceedings of the 2002 IEEE International Symposium on Computer Aided Control System Design (IEEE, 2002), pp. 272–277.

Poland, S. P.

Porcar-Guezenec, R.

Rahman, S. A.

S. A. Rahman and M. J. Booth, “Adaptive optics for high-resolution microscopy: wave front sensing using back scattered light,” Proc. SPIE 8253, 82530I (2012).
[CrossRef]

Rao, C.

Reeds, J. A.

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder–Mead simplex method in low dimensions,” SIAM J. Optim. 9, 112–147 (1998).
[CrossRef]

Ross, T. S.

Rückel, M.

Schitter, G.

Sherman, L.

L. Sherman, J. Y. Ye, O. Albert, and T. B. Norris, “Adaptive correction of depth-induced aberrations in multiphoton scanning microscopy using a deformable mirror,” J. Microsc. 206, 65–71 (2002).
[CrossRef]

O. Albert, L. Sherman, G. Mourou, T. B. Norris, and G. Vdovin, “Smart microscope: an adaptive optics learning system for aberration correction in multiphoton confocal microscopy,” Opt. Lett. 25, 52–54 (2000).
[CrossRef]

So, P. T. C.

J. W. Cha, J. Ballesta, and P. T. C. So, “Shack–Hartmann wavefront-sensor-based adaptive optics system for multiphoton microscopy,” J. Biomed. Opt. 15, 046022 (2010).
[CrossRef]

Soloviev, O.

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Figures (7)

Fig. 1.
Fig. 1.

Schema representing a sensorless adaptive optics system. An unknown aberration applied at the entrance pupil of the system must be corrected by a deformable mirror that is conjugated to the entrance pupil. The measurement y˜(k) made with a photodiode covered by a pinhole is an indicator of the residual aberration in the wavefront. The controller changes control signal u(k) in order to maximize y˜(k).

Fig. 2.
Fig. 2.

Timeline of the iterative aberration correction algorithm. Each iteration consists of a data acquisition part where N+1 data points are acquired and a correction part where correction is performed using Eq. (29) for C time instants. The vectors in the ith data acquisition are taken from a neighborhood of the estimate of x at iteration i1, i.e., x^(i1).

Fig. 3.
Fig. 3.

The spherical wavefront is generated by spatially filtering a laser beam with lens L1 and pinhole P1. The beam is collimated by lens L2 and clipped by iris I1 to fill 10 mm of the aperture of the deformable mirror DM. The membrane of DM is reimaged by lenses L3 and L4 onto a microlens array MLA. C1 and MLA implement a Shack–Hartmann wavefront sensor. Lens L5 focuses the beam onto a photodiode that is covered by a pinhole P2. Flat mirror M1 is used to calibrate the Shack–Hartmann wavefront sensor. An aberration is introduced as an unknown offset x to the control signal of DM. An aberration correction experiment consists of suppressing x when only the measurements of the photodiode are available. Afterward, a measurement of the residual aberration is obtained with the wavefront sensor to assess the performance of the correction.

Fig. 4.
Fig. 4.

Comparison of the experimental computation of matrix Q with Débarre’s method (see Section 3) and our proposed procedure (see Section 3). First a data set of 15246 input–output tuples is acquired (p1=21, p2=13, and p3=21) and Q is computed with Débarre’s method, resulting in Qd. The same input–output data set is used to compute Q with Eq. (16), resulting in Qsdp. The VAFs for Qd and Qsdp are computed over the identification data set. Subsequently, a new input–output data set with 15,000 tuples is acquired for cross validation. In this second set the input aberrations are chosen randomly. The VAFs for Qd and Qsdp are computed using this latter validation set. Such steps are repeated 40 times. (a) Mean value μ, maximum max, minimum min, and standard deviation σ of the identification VAFs for Qd and Qsdp. (b) Mean value μ, maximum max, minimum min, and standard deviation σ of the cross-validation VAFs for Qd and Qsdp. (c) Mean value of Qd and Qsdp over the 40 realizations. The color map is scaled to the maximum and minimum of the elements of Qsdp in order to preserve contrast between the two matrices. Matrix Qd resulted indefinite 32 times out of the 40 trials.

Fig. 5.
Fig. 5.

Optimization (16) was solved for 10 different input–output data sets where the maximum rms of the input aberration (ρmax) is linearly increasing up to 2 rad rms. The VAF is reported for both identification (3750 data points) and validation (1250 data points). Between 0.4 and 0.6 rad rms, the difference between y˜ and Eq. (12) becomes noticeable.

Fig. 6.
Fig. 6.

Each figure reports a summary of the correction of a set of 50 random aberrations. In the upper plot, the mean value, standard deviation, minimum, and maximum of the residual aberrations after the correction are reported in radians. These are denoted, respectively, by a circle, a thick vertical bar, and thin horizontal lines. The same indicators are also reported for the random initial aberrations before correction. (LS1), (LS2), and (LS4) denote respectively 1, 2, and 4 iterations of Eq. (29) as depicted in Fig. 2. (3N) and (5N) are described in Section 4 and [24]. (Simplex) and (SPL) denote the Nelder–Mead simplex method [23]. The horizontal dashed–dotted magenta line denotes a Strehl ratio of 0.9. The lower plots report the mean value of the normalized intensity against sample time for the 50 aberration correction experiments.

Fig. 7.
Fig. 7.

Summary of 50 random aberration correction experiments. The same conventions as in Fig. 6 are employed to report the results. In this case, a Gaussian function instead of Eq. (12) was used to model y˜, as outlined in Section 6. (LS3E) denotes three iterations of Eq. (29) as depicted in Fig. 2.

Equations (29)

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maxu(k)y˜(k),
y˜(k)=(Aλf)2Σ2|Σ1exp[j2πλW(ξ,η,k)j2πλf(ξα+ηβ)]dξdη|2dαdβ+w(k),
y˜(k)(Aλf)2|Σ1exp[j2πλW(ξ,η,k)]dξdη|2+w(k).
y˜(k)(ASλf)2[1(1SΣ1Φ˜(ξ,η,k)2dξdη(1SΣ1Φ˜(ξ,η,k)dξdη)2)]+w(k).
Φ˜(ξ,η,k)=i=1Ψi(ξ,η)vi(k).
Φ˜(ξ,η,k)F(ξ,η)Tv(k).
y˜(k)c˜0v(k)TQ˜v(k)+w(k),
Q˜=c˜0(1SΣ1F(ξ,η)F(ξ,η)Tdξdη(1SΣ1F(ξ,η)dξdη)(1SΣ1F(ξ,η)Tdξdη)).
y˜(k)c˜0(xu(k))TQ˜(xu(k))+w(k).
y(u1,u2,u3)=c˜0[u1u2u3]T[q˜1,1q˜2,1q˜3,1q˜2,1q˜2,2q˜3,2q˜3,1q˜3,2q˜3,3][u1u2u3].
y(u1,u¯)=c˜1[u1u¯]T[q˜1,1q˜2,1q˜2,1q˜2,2][u1u¯].
y(k)=c0+c1T(xu(k))(xu(k))TQ(xu(k))+w(k),
minc0,c1,Qk=1D|y(k)(c0c1Tu(k)u(k)TQu(k))|2s.t.Q0,
minzbAz22,
A=[1u(1)Tu(1)Tu(1)T1u(D)Tu(D)Tu(D)T]RD×(1+N+N2),
minzb1R¯z22s.t.Q0,
y(p)=c0i=1Nλi,i(zipi)2.
maxpiα1pi2+α2pi+α3,
{α1=(y12y2+y3)/(2b2)α2=(y1y3)/(2b)α3=y2.
b(y1y3)/(2y14y2+2y3)
{y(0)=c0λz2y(p¯)=c0λz2+2λp¯zλp¯2.
δy(k,l)=c1T(u(k)u(l))+2(u(k)u(l))TQx+u(l)TQu(l)u(k)TQu(k)+w(k)w(l).
FMx+LMeM=dM,
FM=[2(u(M)u(1))TQ2(u(M)u(M1))TQ]R(M1)×N,
dM=[δy(M,1)+c1T(u(M)u(1))u(1)TQu(1)+u(M)TQu(M)δy(M,M1)+c1T(u(M)u(M1))u(M1)TQu(M1)+u(M)TQu(M)]RM1.
minxeMTeMs.t.FMx+LMeM=dM
(FMTWMFM)x^M=FMTWMdM,
u(N+2)=x^N+112Q1c1.
u(k)=x^k112Q1c1.

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