Abstract

Simulating the turbulence effect on ground telescope observations is of fundamental importance for the design and test of suitable control algorithms for adaptive optics systems. In this paper we propose a multiscale approach for efficiently synthesizing turbulent phases at very high resolution. First, the turbulence is simulated at low resolution, taking advantage of a previously developed method for generating phase screens [J. Opt. Soc. Am. A 25, 515 (2008)]. Then, high-resolution phase screens are obtained as the output of a multiscale linear stochastic system. The multiscale approach significantly improves the computational efficiency of turbulence simulation with respect to recently developed methods [Opt. Express 14, 988 (2006)] [J. Opt. Soc. Am. A 25, 515 (2008)] [J. Opt. Soc. Am. A 25, 463 (2008)]. Furthermore, the proposed procedure ensures good accuracy in reproducing the statistical characteristics of the turbulent phase.

© 2011 Optical Society of America

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References

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  1. F. Roddier, Adaptive Optics in Astronomy (Cambridge University, 1999).
    [CrossRef]
  2. F. Assemat, R. Wilson, and E. Gendron, “Method for simulating infinitely long and non stationary phase screens with optimized memory storage,” Opt. Express 14, 988–999 (2006).
    [CrossRef] [PubMed]
  3. A. Beghi, A. Cenedese, and A. Masiero, “Stochastic realization approach to the efficient simulation of phase screens,” J. Opt. Soc. Am. A 25, 515–525 (2008).
    [CrossRef]
  4. D. Fried and T. Clark, “Extruding Kolmogorov-type phase screen ribbons,” J. Opt. Soc. Am. A 25, 463–468 (2008).
    [CrossRef]
  5. P. S. Maybeck, Stochastic Models, Estimation, and Control (Academic, 1979), Vol.  1.
  6. R. G. Lane and A. Glindemann and J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).
    [CrossRef]
  7. E. Thiebaut and M. Tallon, “Fast minimum variance wavefront reconstruction for extremely large telescope,” J. Opt. Soc. Am. A 27, 1046–1059 (2010).
    [CrossRef]
  8. M. Luettgen, W. Karl, A. Willsky, and R. Tenney, “Multiscale representations of Markov random fields,” IEEE Trans. Signal Process. 41, 3377–3396 (1993).
    [CrossRef]
  9. A. Benveniste, R. Nikoukhah, and A. Willsky, “Multiscale system theory,” IEEE Trans. Circuits Syst. I 41, 2–15 (1994).
    [CrossRef]
  10. K. Daoudi, A. Frakt, and A. Willsky, “Multiscale autoregressive models and wavelets,” IEEE Trans. Inf. Theory 45, 828–845 (1999).
    [CrossRef]
  11. W. Irving and A. Willsky, “A canonical correlations approach to multiscale stochastic realization,” IEEE Trans. Autom. Control 46, 1514–1528 (2001).
    [CrossRef]
  12. A. Frakt and A. Willsky, “Computationally efficient stochastic realization for internal multiscale autoregressive models,” Multidimens. Syst. Signal Process. 12, 109–142 (2001).
    [CrossRef]
  13. I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics 61 (SIAM, 1992).
    [CrossRef]
  14. S. Mallat, A Wavelet Tour of Signal Processing (Academic, 1999).
  15. R. Conan, “Modelisation des effets de l’echelle externe de coherence spatiale du front d’onde pour l’observation a haute resolution angulaire en astronomie,” Ph.D. thesis (Université Nice Sophia Antipolis, 2000).
  16. F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” Prog. Opt. 19, 281–376 (1981).
    [CrossRef]
  17. From Eq. , wi(u) and wi′(u) are practically uncorrelated for large |u−u′|. Furthermore, intuitively, the conditioned value of wi(u), given the local values of xi (i.e., the process at lower resolution), is much more uncorrelated with wi(u′), u≠u′, with respect to wi(u).
  18. J. Woods, “Two-dimensional discrete Markovian fields,” IEEE Trans. Inf. Theory 18, 232–240 (1972).
    [CrossRef]
  19. M. B. Priestley, Spectral Analysis and Time Series (Academic, 1982), Vol.  1.
  20. A. Steinhardt, “Correlation matching by finite length sequences,” IEEE Trans. Acoust. Speech Signal Process. 36, 545–559 (1988).
    [CrossRef]
  21. M. Le Ravalec, B. Noetinger, and L. Hu, “The FFT moving average (FFT-MA) generator: an efficient numerical method for generating and conditioning Gaussian simulations,” Math. Geol. 32, 701–723 (2000).
    [CrossRef]
  22. A. Oppenheim and R. Schafer, Digital Signal Processing(Prentice-Hall, 1975).
  23. Imposing F(θi)(h)=F(ce,i)(h), the solution is restricted to be symmetric. However, there might exist a shorter nonsymmetric sequence of coefficients leading to the same covariances {ce,i(−di),…,ce,i(di)}.
  24. To be more precise, let m¯ and n¯ be the dimensions of a single phase screen (if the wind velocity is parallel to u, then typically m¯=m and n¯≪n) and assume m0≪m and n0≪n; then the number of operations computed by the algorithm to generate an m¯×n¯ phase screen are approximately 64d¯2m¯n¯. When iteratively generating an m×n screen, with n≫n¯, dividing it in approximately n/n¯ phase screens, there is an extra computational load due to the extra computations to ensure the continuity between successive phase screens. Anyway, such extra computational load is usually a minor term in the overall complexity (lower than 10% in our simulations). Similar considerations can be repeated for the memory requirements.
  25. Wavelets different from the Haar transform have better multiscale prediction ability but worse complexity and memory requirements.

2010

2008

2006

2001

W. Irving and A. Willsky, “A canonical correlations approach to multiscale stochastic realization,” IEEE Trans. Autom. Control 46, 1514–1528 (2001).
[CrossRef]

A. Frakt and A. Willsky, “Computationally efficient stochastic realization for internal multiscale autoregressive models,” Multidimens. Syst. Signal Process. 12, 109–142 (2001).
[CrossRef]

2000

R. Conan, “Modelisation des effets de l’echelle externe de coherence spatiale du front d’onde pour l’observation a haute resolution angulaire en astronomie,” Ph.D. thesis (Université Nice Sophia Antipolis, 2000).

M. Le Ravalec, B. Noetinger, and L. Hu, “The FFT moving average (FFT-MA) generator: an efficient numerical method for generating and conditioning Gaussian simulations,” Math. Geol. 32, 701–723 (2000).
[CrossRef]

1999

F. Roddier, Adaptive Optics in Astronomy (Cambridge University, 1999).
[CrossRef]

K. Daoudi, A. Frakt, and A. Willsky, “Multiscale autoregressive models and wavelets,” IEEE Trans. Inf. Theory 45, 828–845 (1999).
[CrossRef]

S. Mallat, A Wavelet Tour of Signal Processing (Academic, 1999).

1994

A. Benveniste, R. Nikoukhah, and A. Willsky, “Multiscale system theory,” IEEE Trans. Circuits Syst. I 41, 2–15 (1994).
[CrossRef]

1993

M. Luettgen, W. Karl, A. Willsky, and R. Tenney, “Multiscale representations of Markov random fields,” IEEE Trans. Signal Process. 41, 3377–3396 (1993).
[CrossRef]

1992

I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics 61 (SIAM, 1992).
[CrossRef]

R. G. Lane and A. Glindemann and J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).
[CrossRef]

1988

A. Steinhardt, “Correlation matching by finite length sequences,” IEEE Trans. Acoust. Speech Signal Process. 36, 545–559 (1988).
[CrossRef]

1982

M. B. Priestley, Spectral Analysis and Time Series (Academic, 1982), Vol.  1.

1981

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” Prog. Opt. 19, 281–376 (1981).
[CrossRef]

1979

P. S. Maybeck, Stochastic Models, Estimation, and Control (Academic, 1979), Vol.  1.

1975

A. Oppenheim and R. Schafer, Digital Signal Processing(Prentice-Hall, 1975).

1972

J. Woods, “Two-dimensional discrete Markovian fields,” IEEE Trans. Inf. Theory 18, 232–240 (1972).
[CrossRef]

Assemat, F.

Beghi, A.

Benveniste, A.

A. Benveniste, R. Nikoukhah, and A. Willsky, “Multiscale system theory,” IEEE Trans. Circuits Syst. I 41, 2–15 (1994).
[CrossRef]

Cenedese, A.

Clark, T.

Conan, R.

R. Conan, “Modelisation des effets de l’echelle externe de coherence spatiale du front d’onde pour l’observation a haute resolution angulaire en astronomie,” Ph.D. thesis (Université Nice Sophia Antipolis, 2000).

Dainty, J. C.

R. G. Lane and A. Glindemann and J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).
[CrossRef]

Daoudi, K.

K. Daoudi, A. Frakt, and A. Willsky, “Multiscale autoregressive models and wavelets,” IEEE Trans. Inf. Theory 45, 828–845 (1999).
[CrossRef]

Daubechies, I.

I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics 61 (SIAM, 1992).
[CrossRef]

Frakt, A.

A. Frakt and A. Willsky, “Computationally efficient stochastic realization for internal multiscale autoregressive models,” Multidimens. Syst. Signal Process. 12, 109–142 (2001).
[CrossRef]

K. Daoudi, A. Frakt, and A. Willsky, “Multiscale autoregressive models and wavelets,” IEEE Trans. Inf. Theory 45, 828–845 (1999).
[CrossRef]

Fried, D.

Gendron, E.

Glindemann, A.

R. G. Lane and A. Glindemann and J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).
[CrossRef]

Hu, L.

M. Le Ravalec, B. Noetinger, and L. Hu, “The FFT moving average (FFT-MA) generator: an efficient numerical method for generating and conditioning Gaussian simulations,” Math. Geol. 32, 701–723 (2000).
[CrossRef]

Irving, W.

W. Irving and A. Willsky, “A canonical correlations approach to multiscale stochastic realization,” IEEE Trans. Autom. Control 46, 1514–1528 (2001).
[CrossRef]

Karl, W.

M. Luettgen, W. Karl, A. Willsky, and R. Tenney, “Multiscale representations of Markov random fields,” IEEE Trans. Signal Process. 41, 3377–3396 (1993).
[CrossRef]

Lane, R. G.

R. G. Lane and A. Glindemann and J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).
[CrossRef]

Le Ravalec, M.

M. Le Ravalec, B. Noetinger, and L. Hu, “The FFT moving average (FFT-MA) generator: an efficient numerical method for generating and conditioning Gaussian simulations,” Math. Geol. 32, 701–723 (2000).
[CrossRef]

Luettgen, M.

M. Luettgen, W. Karl, A. Willsky, and R. Tenney, “Multiscale representations of Markov random fields,” IEEE Trans. Signal Process. 41, 3377–3396 (1993).
[CrossRef]

Mallat, S.

S. Mallat, A Wavelet Tour of Signal Processing (Academic, 1999).

Masiero, A.

Maybeck, P. S.

P. S. Maybeck, Stochastic Models, Estimation, and Control (Academic, 1979), Vol.  1.

Nikoukhah, R.

A. Benveniste, R. Nikoukhah, and A. Willsky, “Multiscale system theory,” IEEE Trans. Circuits Syst. I 41, 2–15 (1994).
[CrossRef]

Noetinger, B.

M. Le Ravalec, B. Noetinger, and L. Hu, “The FFT moving average (FFT-MA) generator: an efficient numerical method for generating and conditioning Gaussian simulations,” Math. Geol. 32, 701–723 (2000).
[CrossRef]

Oppenheim, A.

A. Oppenheim and R. Schafer, Digital Signal Processing(Prentice-Hall, 1975).

Priestley, M. B.

M. B. Priestley, Spectral Analysis and Time Series (Academic, 1982), Vol.  1.

Roddier, F.

F. Roddier, Adaptive Optics in Astronomy (Cambridge University, 1999).
[CrossRef]

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” Prog. Opt. 19, 281–376 (1981).
[CrossRef]

Schafer, R.

A. Oppenheim and R. Schafer, Digital Signal Processing(Prentice-Hall, 1975).

Steinhardt, A.

A. Steinhardt, “Correlation matching by finite length sequences,” IEEE Trans. Acoust. Speech Signal Process. 36, 545–559 (1988).
[CrossRef]

Tallon, M.

Tenney, R.

M. Luettgen, W. Karl, A. Willsky, and R. Tenney, “Multiscale representations of Markov random fields,” IEEE Trans. Signal Process. 41, 3377–3396 (1993).
[CrossRef]

Thiebaut, E.

Willsky, A.

W. Irving and A. Willsky, “A canonical correlations approach to multiscale stochastic realization,” IEEE Trans. Autom. Control 46, 1514–1528 (2001).
[CrossRef]

A. Frakt and A. Willsky, “Computationally efficient stochastic realization for internal multiscale autoregressive models,” Multidimens. Syst. Signal Process. 12, 109–142 (2001).
[CrossRef]

K. Daoudi, A. Frakt, and A. Willsky, “Multiscale autoregressive models and wavelets,” IEEE Trans. Inf. Theory 45, 828–845 (1999).
[CrossRef]

A. Benveniste, R. Nikoukhah, and A. Willsky, “Multiscale system theory,” IEEE Trans. Circuits Syst. I 41, 2–15 (1994).
[CrossRef]

M. Luettgen, W. Karl, A. Willsky, and R. Tenney, “Multiscale representations of Markov random fields,” IEEE Trans. Signal Process. 41, 3377–3396 (1993).
[CrossRef]

Wilson, R.

Woods, J.

J. Woods, “Two-dimensional discrete Markovian fields,” IEEE Trans. Inf. Theory 18, 232–240 (1972).
[CrossRef]

IEEE Trans. Acoust. Speech Signal Process.

A. Steinhardt, “Correlation matching by finite length sequences,” IEEE Trans. Acoust. Speech Signal Process. 36, 545–559 (1988).
[CrossRef]

IEEE Trans. Autom. Control

W. Irving and A. Willsky, “A canonical correlations approach to multiscale stochastic realization,” IEEE Trans. Autom. Control 46, 1514–1528 (2001).
[CrossRef]

IEEE Trans. Circuits Syst. I

A. Benveniste, R. Nikoukhah, and A. Willsky, “Multiscale system theory,” IEEE Trans. Circuits Syst. I 41, 2–15 (1994).
[CrossRef]

IEEE Trans. Inf. Theory

K. Daoudi, A. Frakt, and A. Willsky, “Multiscale autoregressive models and wavelets,” IEEE Trans. Inf. Theory 45, 828–845 (1999).
[CrossRef]

J. Woods, “Two-dimensional discrete Markovian fields,” IEEE Trans. Inf. Theory 18, 232–240 (1972).
[CrossRef]

IEEE Trans. Signal Process.

M. Luettgen, W. Karl, A. Willsky, and R. Tenney, “Multiscale representations of Markov random fields,” IEEE Trans. Signal Process. 41, 3377–3396 (1993).
[CrossRef]

J. Opt. Soc. Am. A

Math. Geol.

M. Le Ravalec, B. Noetinger, and L. Hu, “The FFT moving average (FFT-MA) generator: an efficient numerical method for generating and conditioning Gaussian simulations,” Math. Geol. 32, 701–723 (2000).
[CrossRef]

Multidimens. Syst. Signal Process.

A. Frakt and A. Willsky, “Computationally efficient stochastic realization for internal multiscale autoregressive models,” Multidimens. Syst. Signal Process. 12, 109–142 (2001).
[CrossRef]

Opt. Express

Prog. Opt.

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” Prog. Opt. 19, 281–376 (1981).
[CrossRef]

Waves Random Media

R. G. Lane and A. Glindemann and J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).
[CrossRef]

Other

F. Roddier, Adaptive Optics in Astronomy (Cambridge University, 1999).
[CrossRef]

P. S. Maybeck, Stochastic Models, Estimation, and Control (Academic, 1979), Vol.  1.

From Eq. , wi(u) and wi′(u) are practically uncorrelated for large |u−u′|. Furthermore, intuitively, the conditioned value of wi(u), given the local values of xi (i.e., the process at lower resolution), is much more uncorrelated with wi(u′), u≠u′, with respect to wi(u).

M. B. Priestley, Spectral Analysis and Time Series (Academic, 1982), Vol.  1.

I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics 61 (SIAM, 1992).
[CrossRef]

S. Mallat, A Wavelet Tour of Signal Processing (Academic, 1999).

R. Conan, “Modelisation des effets de l’echelle externe de coherence spatiale du front d’onde pour l’observation a haute resolution angulaire en astronomie,” Ph.D. thesis (Université Nice Sophia Antipolis, 2000).

A. Oppenheim and R. Schafer, Digital Signal Processing(Prentice-Hall, 1975).

Imposing F(θi)(h)=F(ce,i)(h), the solution is restricted to be symmetric. However, there might exist a shorter nonsymmetric sequence of coefficients leading to the same covariances {ce,i(−di),…,ce,i(di)}.

To be more precise, let m¯ and n¯ be the dimensions of a single phase screen (if the wind velocity is parallel to u, then typically m¯=m and n¯≪n) and assume m0≪m and n0≪n; then the number of operations computed by the algorithm to generate an m¯×n¯ phase screen are approximately 64d¯2m¯n¯. When iteratively generating an m×n screen, with n≫n¯, dividing it in approximately n/n¯ phase screens, there is an extra computational load due to the extra computations to ensure the continuity between successive phase screens. Anyway, such extra computational load is usually a minor term in the overall complexity (lower than 10% in our simulations). Similar considerations can be repeated for the memory requirements.

Wavelets different from the Haar transform have better multiscale prediction ability but worse complexity and memory requirements.

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Figures (6)

Fig. 1
Fig. 1

(a) Coordinates on the telescope image domain. (b) Two points, ( u , v ) and ( u , v ) , separated by a distance ρ on the telescope aperture plane. (c) Example of phase screen.

Fig. 2
Fig. 2

Haar wavelet decomposition of a phase screen. i is the scale index, and M = 2 (number of scales). Conventionally i = M corresponds to the high-resolution turbulent phase, while in the wavelet decomposition i ranges between 0 and M 1 . The figures of the ith row represent from left to right the low-pass version of the current phase screen ( x i ) and the details on the horizontal ( w i h ), vertical ( w i v ), and diagonal ( w i d ) direction at scale i.

Fig. 3
Fig. 3

Phase screen synthesis: comparison of low- ( x 0 l , top) and high-resolution ( x 6 l , bottom) simulated turbulent phase on 8 m × 8 m (left) and 2.1 m × 2.1 m (right) windows. Resolutions in pixels are 60 × 60 (top left) and 16 × 16 (top right) pixels, 3840 × 3840 (bottom left) and 1024 × 1024 (bottom right) pixels.

Fig. 4
Fig. 4

Comparison of the theoretical structure function (dashed curve) with the sample structure function (solid curve) evaluated along the wind direction computed from a 8 m × 10698 m phase screen. The values of the parameters are set to L 0 = 50 m , r 0 = 0.2 m , d = 8 m , p s = 0.0021 m .

Fig. 5
Fig. 5

Comparison of the theoretical structure function (dashed curve) with the sample structure function (solid curve) evaluated along the wind direction computed from a 320 m × 80080 m phase screen. The values of the parameters are set to L 0 = 50 m , r 0 = 0.2 m , d = 42 m , p s = 0.031 m .

Fig. 6
Fig. 6

Computational complexity of turbulent phase synthesis: comparison between dynamic model (5) used as in [2, 3, 4] [solid curve, O ( m 2 n ) ] and procedure of Section 3 [dotted–dashed line, O ( m n ) ]. n = 1000 ; m varies between 40 and 4000.

Equations (33)

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

c ϕ ( ρ ) = ( L 0 r 0 ) 5 / 3 η 2 ( 2 π ρ L 0 ) 5 / 6 K 5 / 6 ( 2 π ρ L 0 ) ,
ϕ ( u , v , t ) = l = 1 L γ l ψ l ( u , v , t ) ,
c ψ l ( ρ ) = c ϕ ( ρ ) .
E [ ψ l ( u , v , t ) ψ l ( u , v , t ) ] = 0 , l l .
ψ l ( u , v , t + k T s ) = ψ l ( u ν l , u k T s , v ν l , v k T s , t ) ,
{ χ ( u + 1 ) = A 0 χ ( u ) + K 0 ξ ( u ) φ ( u ) = C 0 χ ( u ) + ξ ( u ) ,
[ x i ( u , v ) w i h ( u , v ) w i v ( u , v ) w i d ( u , v ) ] = C [ x i + 1 ( 2 u , 2 v ) x i + 1 ( 2 u , 2 v + 1 ) x i + 1 ( 2 u + 1 , 2 v ) x i + 1 ( 2 u + 1 , 2 v + 1 ) ] ,
[ x i + 1 ( 2 u , 2 v ) x i + 1 ( 2 u , 2 v + 1 ) x i + 1 ( 2 u + 1 , 2 v ) x i + 1 ( 2 u + 1 , 2 v + 1 ) ] = C [ x i ( u , v ) w i h ( u , v ) w i v ( u , v ) w i d ( u , v ) ] ,
C = 1 2 [ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ] ,
E [ x M ( u , v ) ] = 0 ,     ( u , v ) ,
c x , M ( u , v ) = E [ x M ( u , v ) x M ( 0 , 0 ) ] = c ϕ ( | ( u , v ) | ) ,     ( u , v ) ,
E [ x i ( u , v ) w i h ( u , v ) w i v ( u , v ) w i d ( u , v ) ] = [ 0 0 0 0 ] ,     ( u , v ) , 0 i < M ,
E [ [ x i 1 ( u , v ) w i 1 h ( u , v ) w i 1 v ( u , v ) w i 1 d ( u , v ) ] [ x i 1 ( 0 , 0 ) w i 1 h ( 0 , 0 ) w i 1 v ( 0 , 0 ) w i 1 d ( 0 , 0 ) ] ] = C [ c x , i ( u , v ) c x , i ( u , v 1 ) c x , i ( u 1 , v ) c x , i ( u 1 , v 1 ) c x , i ( u , v + 1 ) c x , i ( u , v ) c x , i ( u 1 , v + 1 ) c x , i ( u 1 , v ) c x , i ( u + 1 , v ) c x , i ( u + 1 , v 1 ) c x , i ( u , v ) c x , i ( u , v 1 ) c x , i ( u + 1 , v + 1 ) c x , i ( u + 1 , v ) c x , i ( u , v + 1 ) c x , i ( u , v ) ] C ,
[ x i ( u ) w i ( u ) ] = C 1 [ x i + 1 ( 2 u ) x i + 1 ( 2 u + 1 ) ] , u L ,
C 1 = 1 2 [ 1 1 1 1 ] .
c x , M ( u ) = E [ x M ( u ) x M ( 0 ) ] = c ϕ ( | u | ) , u L ,
E [ x i 1 ( u ) w i 1 ( u ) ] [ x i 1 ( 0 ) w i 1 ( 0 ) ] = C 1 [ c x , i ( 2 u ) c x , i ( 2 u 1 ) c x , i ( 2 u + 1 ) c x , i ( 2 u ) ] C 1 ,
c x , i ( u ) 0 , c w , i ( u ) 0 , c x w , i ( u ) 0 , for large     | u | ,
p ( w i ( u ) | x i ( u ) , u L ; w i ( u ) , u N i ( u ) ) = p ( w i ( u ) | x i ( u ) , u N i ( u ) ) , u L ,
N i ( u ) = { u L | 0 | u u | d ¯ i } ,
w i ( u ) = u N i ( u ) a i , | u u | x i ( u ) + e i ( u ) = w ^ i ( u ) + e i ( u ) , u L ,
w ^ i ( u ) = u N i ( u ) a i , | u u | x i ( u )
e i ( u ) = k = + θ i ( u k ) ϵ i ( k ) ,
E [ e i ( u ) e i ( u ) ] = E [ e i ( u ) ( w i ( u ) w ^ i ( u ) ) ] = E [ e i ( u ) w i ( u ) ] = 0 ,     u such that     | u u | > d i ,
c e , i ( u ) = E [ e i ( u ) e i ( 0 ) ] = E [ w i ( u ) w i ( 0 ) ] u N i ( u ) a i , | u u | E [ x i ( u ) w i ( 0 ) ] u N i ( 0 ) a i , | u | E [ w i ( u ) x i ( u ) ] + u N i ( 0 ) a i , | u | u N i ( u ) a i , | u u | c x , i ( u u ) ,
e i ( u ) = k = d i d i θ i ( k ) ϵ i ( u k ) .
c e , i ( k ) = ( θ i * θ i ) ( k ) ,
w i ( u ) = u N i ( u ) a i , | u u | x i ( u ) + u N i ( u ) θ i ( u u ) ϵ i ( u ) , u L .
F ( c e , i ) ( h ) = F ( θ i * θ i ) ( h ) = F ( θ i ) ( h ) · F ( θ i ) ( h ) .
F ( c e , i ) ( h ) = F ( θ i ) ( h ) · ( F ( θ i ) ( h ) ) * 0 ,
w i ( u ) = w ^ i ( u ) + e i ( u ) , u = u i , 1 u i , 2 ,
E [ e i ( u ) e i ( u + δ i ) ] 0 ;
D ϕ ( ρ ) = 2 ( c ϕ ( 0 ) c ϕ ( ρ ) ) .

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