Abstract

The question of just how much to defocus the second image in the conventional two-channel phase-diversity speckle imaging technique may be addressed in a number of ways. Fisher information furnishes a useful metric for optimizing the choice of defocus as a functional of the object class, operating conditions, and the imaging task. Approximate closed-form expressions for the Fisher information relative to object parameters, rather than the pupil phase, are derived and discussed for phase-diversity-speckle imaging under conditions of strong turbulence and additive Gaussian noise. As an application of our general information-theoretic approach, the optimization of defocus when the imaging task is to estimate the midfrequency power spectrum of the object is discussed.

© 2004 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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2003 (4)

2002 (1)

T. Torgersen, D. W. Tyler, “Practical considerations in restoring images from phase-diverse speckle data,” Astron. Soc. Pac. Conf. Ser. 114, 671–685 (2002).
[CrossRef]

2000 (1)

1999 (1)

1995 (1)

1993 (1)

1992 (2)

P. Jakobsen, P. Greenfield, R. Jedrzejewski, “The Cramér–Rao lower bound and stellar photometry with aberrated HST images,” Astron. Astrophys. 253, 329–332 (1992).

R. G. Paxman, T. J. Schulz, J. R. Fienup, “Joint estimation of object and aberrations by using phase diversity,” J. Opt. Soc. Am. A 9, 1072–1085 (1992).
[CrossRef]

1989 (1)

1988 (1)

T. V. Arak, A. Yu. Zaitsev, “Uniform limit theorems for sums of independent random variables,” Proc. Steklov Inst. Math. 174, 1–222 (1988).

1987 (1)

1986 (1)

J. N. Cederquist, S. R. Robinson, D. Kryskowski, J. R. Fienup, C. C. Wackerman, “Cramér–Rao lower bound on wavefront sensor error,” Opt. Eng. 25, 586–592 (1986).
[CrossRef]

1982 (1)

R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21, 829–832 (1982).
[CrossRef]

1980 (1)

P. Hall, “Characterizing the rate of convergence in the central limit theorem,” Ann. Prob. 8, 1037–1048 (1980).
[CrossRef]

Arak, T. V.

T. V. Arak, A. Yu. Zaitsev, “Uniform limit theorems for sums of independent random variables,” Proc. Steklov Inst. Math. 174, 1–222 (1988).

Blanc, A.

Bowers, C. W.

Bresler, Y.

J. C. Ye, Y. Bresler, P. Moulin, “Cramér–Rao bounds for parametric shape estimation in inverse problems,” IEEE Trans. Image Process. 12, 71–84 (2003).
[CrossRef]

Carrara, D. A.

J. R. Fienup, B. J. Thelen, R. G. Paxman, D. A. Carrara, “Comparison of phase diversity and curvature wavefront sensing,” in Adaptive Optical System Technologies, D. Bonaccini, R. K. Tyson, eds., Proc. SPIE3353, 930–940 (1998).
[CrossRef]

Cathey, W. T.

Cederquist, J. N.

Chan, T.

C. R. Vogel, T. Chan, R. J. Plemmons, “Fast algorithms for phase-diversity-based blind deconvolution,” in Adaptive Optical System Technologies, D. Bonaccini, R. K. Tyson, eds., Proc. SPIE3353, 994–1005 (1998).
[CrossRef]

Chaudhuri, S.

Chellappa, R.

Cramér, H.

H. Cramér, Random Variables and Probability Distributions, 3rd ed. (Cambridge U. Press, Cambridge, UK, 1970).

Dean, B. H.

Dowski, E. R.

Fienup, J. R.

J. R. Fienup, J. C. Marron, T. J. Schulz, J. H. Seldin, “Hubble Space Telescope characterized by using phase retrieval algorithms,” Appl. Opt. 32, 1747–1767 (1993).
[CrossRef] [PubMed]

R. G. Paxman, T. J. Schulz, J. R. Fienup, “Joint estimation of object and aberrations by using phase diversity,” J. Opt. Soc. Am. A 9, 1072–1085 (1992).
[CrossRef]

J. N. Cederquist, J. R. Fienup, C. C. Wackerman, S. R. Robinson, D. Kryskowski, “Wave-front estimation from Fourier intensity measurements,” J. Opt. Soc. Am. A 6, 1020–1026 (1989).
[CrossRef]

J. N. Cederquist, S. R. Robinson, D. Kryskowski, J. R. Fienup, C. C. Wackerman, “Cramér–Rao lower bound on wavefront sensor error,” Opt. Eng. 25, 586–592 (1986).
[CrossRef]

J. R. Fienup, B. J. Thelen, R. G. Paxman, D. A. Carrara, “Comparison of phase diversity and curvature wavefront sensing,” in Adaptive Optical System Technologies, D. Bonaccini, R. K. Tyson, eds., Proc. SPIE3353, 930–940 (1998).
[CrossRef]

Gerwe, D. R.

Gonsalves, R. A.

R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21, 829–832 (1982).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

Greenfield, P.

P. Jakobsen, P. Greenfield, R. Jedrzejewski, “The Cramér–Rao lower bound and stellar photometry with aberrated HST images,” Astron. Astrophys. 253, 329–332 (1992).

Hall, P.

P. Hall, “Characterizing the rate of convergence in the central limit theorem,” Ann. Prob. 8, 1037–1048 (1980).
[CrossRef]

Hill, J. L.

Idell, P. S.

Idier, J.

Jakobsen, P.

P. Jakobsen, P. Greenfield, R. Jedrzejewski, “The Cramér–Rao lower bound and stellar photometry with aberrated HST images,” Astron. Astrophys. 253, 329–332 (1992).

Jedrzejewski, R.

P. Jakobsen, P. Greenfield, R. Jedrzejewski, “The Cramér–Rao lower bound and stellar photometry with aberrated HST images,” Astron. Astrophys. 253, 329–332 (1992).

Kay, S. M.

S. M. Kay, Fundamentals of Statistical Signal Processing: I. Estimation Theory (Prentice Hall, Englewood Cliffs, N.J., 1993).

Kryskowski, D.

J. N. Cederquist, J. R. Fienup, C. C. Wackerman, S. R. Robinson, D. Kryskowski, “Wave-front estimation from Fourier intensity measurements,” J. Opt. Soc. Am. A 6, 1020–1026 (1989).
[CrossRef]

J. N. Cederquist, S. R. Robinson, D. Kryskowski, J. R. Fienup, C. C. Wackerman, “Cramér–Rao lower bound on wavefront sensor error,” Opt. Eng. 25, 586–592 (1986).
[CrossRef]

Lee, D. J.

Marron, J. C.

Moulin, P.

J. C. Ye, Y. Bresler, P. Moulin, “Cramér–Rao bounds for parametric shape estimation in inverse problems,” IEEE Trans. Image Process. 12, 71–84 (2003).
[CrossRef]

Mugnier, L.

Pauca, V. P.

S. Prasad, T. Torgersen, V. P. Pauca, R. Plemmons, J. van der Gracht, “Engineering the pupil phase to improve image quality,” in Visual Information Processing XII, Z. Rahman, R. Schowengerdt, S. Reichenbach, eds., Proc. SPIE5108, 1–12 (2003).
[CrossRef]

Paxman, R. G.

R. G. Paxman, T. J. Schulz, J. R. Fienup, “Joint estimation of object and aberrations by using phase diversity,” J. Opt. Soc. Am. A 9, 1072–1085 (1992).
[CrossRef]

J. R. Fienup, B. J. Thelen, R. G. Paxman, D. A. Carrara, “Comparison of phase diversity and curvature wavefront sensing,” in Adaptive Optical System Technologies, D. Bonaccini, R. K. Tyson, eds., Proc. SPIE3353, 930–940 (1998).
[CrossRef]

Plemmons, R.

S. Prasad, T. Torgersen, V. P. Pauca, R. Plemmons, J. van der Gracht, “Engineering the pupil phase to improve image quality,” in Visual Information Processing XII, Z. Rahman, R. Schowengerdt, S. Reichenbach, eds., Proc. SPIE5108, 1–12 (2003).
[CrossRef]

Plemmons, R. J.

C. R. Vogel, T. Chan, R. J. Plemmons, “Fast algorithms for phase-diversity-based blind deconvolution,” in Adaptive Optical System Technologies, D. Bonaccini, R. K. Tyson, eds., Proc. SPIE3353, 994–1005 (1998).
[CrossRef]

Prasad, S.

S. Prasad, T. Torgersen, V. P. Pauca, R. Plemmons, J. van der Gracht, “Engineering the pupil phase to improve image quality,” in Visual Information Processing XII, Z. Rahman, R. Schowengerdt, S. Reichenbach, eds., Proc. SPIE5108, 1–12 (2003).
[CrossRef]

Rajagopalan, A. N.

Robinson, S. R.

J. N. Cederquist, J. R. Fienup, C. C. Wackerman, S. R. Robinson, D. Kryskowski, “Wave-front estimation from Fourier intensity measurements,” J. Opt. Soc. Am. A 6, 1020–1026 (1989).
[CrossRef]

J. N. Cederquist, S. R. Robinson, D. Kryskowski, J. R. Fienup, C. C. Wackerman, “Cramér–Rao lower bound on wavefront sensor error,” Opt. Eng. 25, 586–592 (1986).
[CrossRef]

Roggemann, M. C.

Rosenblatt, M.

M. Rosenblatt, Stationary Sequences and Random Fields (Birkhäuser, Boston, Mass., 1985).

Schulz, T. J.

Seldin, J. H.

Thelen, B. J.

J. R. Fienup, B. J. Thelen, R. G. Paxman, D. A. Carrara, “Comparison of phase diversity and curvature wavefront sensing,” in Adaptive Optical System Technologies, D. Bonaccini, R. K. Tyson, eds., Proc. SPIE3353, 930–940 (1998).
[CrossRef]

Torgersen, T.

T. Torgersen, D. W. Tyler, “Practical considerations in restoring images from phase-diverse speckle data,” Astron. Soc. Pac. Conf. Ser. 114, 671–685 (2002).
[CrossRef]

S. Prasad, T. Torgersen, V. P. Pauca, R. Plemmons, J. van der Gracht, “Engineering the pupil phase to improve image quality,” in Visual Information Processing XII, Z. Rahman, R. Schowengerdt, S. Reichenbach, eds., Proc. SPIE5108, 1–12 (2003).
[CrossRef]

Tyler, D. W.

T. Torgersen, D. W. Tyler, “Practical considerations in restoring images from phase-diverse speckle data,” Astron. Soc. Pac. Conf. Ser. 114, 671–685 (2002).
[CrossRef]

van der Gracht, J.

S. Prasad, T. Torgersen, V. P. Pauca, R. Plemmons, J. van der Gracht, “Engineering the pupil phase to improve image quality,” in Visual Information Processing XII, Z. Rahman, R. Schowengerdt, S. Reichenbach, eds., Proc. SPIE5108, 1–12 (2003).
[CrossRef]

Van Trees, H.

H. Van Trees, Detection, Estimation, and Modulation Theory (Wiley, New York, 1968).

Vogel, C. R.

C. R. Vogel, T. Chan, R. J. Plemmons, “Fast algorithms for phase-diversity-based blind deconvolution,” in Adaptive Optical System Technologies, D. Bonaccini, R. K. Tyson, eds., Proc. SPIE3353, 994–1005 (1998).
[CrossRef]

Wackerman, C. C.

Welsh, B. M.

Ye, J. C.

J. C. Ye, Y. Bresler, P. Moulin, “Cramér–Rao bounds for parametric shape estimation in inverse problems,” IEEE Trans. Image Process. 12, 71–84 (2003).
[CrossRef]

Zaitsev, A. Yu.

T. V. Arak, A. Yu. Zaitsev, “Uniform limit theorems for sums of independent random variables,” Proc. Steklov Inst. Math. 174, 1–222 (1988).

Ann. Prob. (1)

P. Hall, “Characterizing the rate of convergence in the central limit theorem,” Ann. Prob. 8, 1037–1048 (1980).
[CrossRef]

Appl. Opt. (2)

Astron. Astrophys. (1)

P. Jakobsen, P. Greenfield, R. Jedrzejewski, “The Cramér–Rao lower bound and stellar photometry with aberrated HST images,” Astron. Astrophys. 253, 329–332 (1992).

Astron. Soc. Pac. Conf. Ser. (1)

T. Torgersen, D. W. Tyler, “Practical considerations in restoring images from phase-diverse speckle data,” Astron. Soc. Pac. Conf. Ser. 114, 671–685 (2002).
[CrossRef]

IEEE Trans. Image Process. (1)

J. C. Ye, Y. Bresler, P. Moulin, “Cramér–Rao bounds for parametric shape estimation in inverse problems,” IEEE Trans. Image Process. 12, 71–84 (2003).
[CrossRef]

J. Opt. Soc. Am. A (8)

Opt. Eng. (2)

R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21, 829–832 (1982).
[CrossRef]

J. N. Cederquist, S. R. Robinson, D. Kryskowski, J. R. Fienup, C. C. Wackerman, “Cramér–Rao lower bound on wavefront sensor error,” Opt. Eng. 25, 586–592 (1986).
[CrossRef]

Proc. Steklov Inst. Math. (1)

T. V. Arak, A. Yu. Zaitsev, “Uniform limit theorems for sums of independent random variables,” Proc. Steklov Inst. Math. 174, 1–222 (1988).

Other (9)

M. Rosenblatt, Stationary Sequences and Random Fields (Birkhäuser, Boston, Mass., 1985).

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

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

S. Prasad, T. Torgersen, V. P. Pauca, R. Plemmons, J. van der Gracht, “Engineering the pupil phase to improve image quality,” in Visual Information Processing XII, Z. Rahman, R. Schowengerdt, S. Reichenbach, eds., Proc. SPIE5108, 1–12 (2003).
[CrossRef]

H. Cramér, Random Variables and Probability Distributions, 3rd ed. (Cambridge U. Press, Cambridge, UK, 1970).

J. R. Fienup, B. J. Thelen, R. G. Paxman, D. A. Carrara, “Comparison of phase diversity and curvature wavefront sensing,” in Adaptive Optical System Technologies, D. Bonaccini, R. K. Tyson, eds., Proc. SPIE3353, 930–940 (1998).
[CrossRef]

C. R. Vogel, T. Chan, R. J. Plemmons, “Fast algorithms for phase-diversity-based blind deconvolution,” in Adaptive Optical System Technologies, D. Bonaccini, R. K. Tyson, eds., Proc. SPIE3353, 994–1005 (1998).
[CrossRef]

H. Van Trees, Detection, Estimation, and Modulation Theory (Wiley, New York, 1968).

S. M. Kay, Fundamentals of Statistical Signal Processing: I. Estimation Theory (Prentice Hall, Englewood Cliffs, N.J., 1993).

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Figures (6)

Fig. 1
Fig. 1

OTF covariances in the two images for D/r0=20 plotted as functions of the spatial frequency, in units of D/(2λF), along the x axis for (a) τ˜=5 and (b) τ˜=10.

Fig. 2
Fig. 2

Surface plots of joint FI as functions of the spatial frequency u along the x axis and the defocus parameter τ˜ for SNR at dc at 100, equal detector noise in the two channels, and (a) D/r0=10, and (b) D/r0=20.

Fig. 3
Fig. 3

Fractional (a) increase in the integrated FI, and (b) decrease in the integrated CRLB due to the defocused image, each plotted as a function of the defocus parameter τ˜ for D/r0=10, SNR at dc at 100 and three different values of detector–noise ratio η: 0.1, 1, and 10.

Fig. 4
Fig. 4

Same as Fig. 3 except D/r0=20.

Fig. 5
Fig. 5

Fractional (a) increase in the integrated FI and (b) decrease in the integrated CRLB due to the defocused image, each plotted as a function of the defocus parameter τ˜ for D/r0=10, detector–noise ratio η=1, and three different values of SNR at dc: 10, 100, and 1000.

Fig. 6
Fig. 6

Same as Fig. 5 except D/r0=20.

Equations (88)

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

h˜0(u)=1AP(ρ)P(ρ-uλF)×exp{i[ϕ(ρ)-ϕ(ρ-uλF)]}d2ρ,
gi(x)=f(x-x)hi(x)dx+ni(x),i=0, 1,
ni(x1)nj(x2)=σi2δijδ(x1-x2).
g˜i(u)=f˜(u)h˜i(u)+n˜i(u),
z˜(u)=z(x)exp(-i2πux)dx
Δg˜i(u)Δg˜j*(u)=c˜ij(g)(u)δ(u-u),
c˜ij(g)(u)=c˜ji(g)(-u)=c˜ji(g)*(u).
P˜(0,1)(g˜0, g˜1)u1det c˜(g)(u)1/2×exp-12 i,jΔg˜i*(u)k˜ij(g)(u)×Δg˜j(u)du,
jc˜ij(g)(u)k˜jl(g)(u)=δil
k˜ij(g)(u)=(-1)i+jc˜j¯i¯(g)(u)det c˜(g)(u),
P˜(0)(g˜0)u1c˜00(g)(u)1/2×exp-12 du|Δg˜0(u)|2/c˜00(g)(u).
g˜i(u)=f˜(u)h˜i(u),
c˜ij(g)(u)= |f˜(u)|2c˜ij(h)(u)+σi2δij.
Jij(θ)= ln P(d|θ)θi*  ln P(d|θ)θj,
 ln Pθxδ ln Pδθ(x),
limΔx01Δx ln Pθx=δ ln Pδθ(x).
ln P˜(0)(g˜0)=χ0-12du  ln c˜00(g)(u)du-12 |Δg˜0(u)|2/c˜00(g)(u)du,
δ ln P˜(0)(g˜0)δf˜(u1)=-12du  1c˜00(g)(u) δc˜00(g)(u)δf(u1) du+Δg˜0*(u1)h˜0(u1)c˜00(g)(u1)+12  |Δg˜0(u)|2|c˜00(g)(u)|2 δc˜00(g)(u)δf(u1) du,
δ ln P˜(0)(g˜0)δf˜(u1)=Δg˜0*(u1)h˜0(u1)c˜00(g)(u1)+12  1[c˜00(g)(u)]2 δc˜00(g)(u)δf(u1) [|Δg˜0(u)|2-c˜00(g)(u)δ(0)]du.
J(0)[f˜(u1), f˜(u2)]
=δ ln P˜(0)(g˜0)δf˜*(u1)δ ln P˜(0)(g˜0)δf˜(u2)=h˜0*(u1)h˜0(u2)c˜00(g)(u1)c˜00(g)(u2) Δg˜0(u1)Δg˜0*(u2)+14 du1du2×1[c˜00(g)(u1)]2[c˜00(g)(u2)]2 δc˜00(g)(u1)δf*(u1) δc˜00(g)(u2)δf(u2)×[|Δg˜0(u1)|2-c˜00(g)(u1)δ(0)][|Δg˜0(u2)|2-c˜00(g)(u2)δ(0)].
Δg˜i(ui)Δg˜j*(uj)Δg˜k(uk)Δg˜l*(ul)=Δg˜i(ui)Δg˜j*(uj)Δg˜k(uk)Δg˜l*(ul)+Δg˜i(ui)Δg˜k(uk)Δg˜j*(uj)Δg˜l*(ul)+Δg˜i(ui)Δg˜l*(ul)Δg˜k(uk)Δg˜j*(uj),
J(0)[f˜(u1), f˜(u2)]
=|h˜0(u1)|2c˜00(g)(u1) δ(u1-u2)+14 du1du2×1[c˜00(g)(u1)]2[c˜00(g)(u2)]2 δc˜00(g)(u1)δf*(u1) δc˜00(g)(u2)δf(u2)×{[c˜00(g)(u1)]2[δ(u1-u2)]2+[c˜00(g)(u1)]2[δ(u1+u2)]2}.
J(0)[f˜(u1), f˜(u2)]
=|h˜0(u1)|2c˜00(g)(u1) δ(u1-u2)+δ(0)2 du 1[c˜00(g)(u)]2 δc˜00(g)(u)δf*(u1) δc˜00(g)(u)δf(u2),
ln P˜(0,1)(g˜0, g˜1)=χ01-12du  ln det c˜(g)(u)du-12 i,jΔg˜i*(u)k˜ij(g)(u)Δg˜j(u)du,
δ ln P˜(0,1)(g˜0, g˜1)δf˜(u1)=i,jΔg˜i*(u1)k˜ij(g)(u1)h˜j(u1)-12 i,j[Δg˜i*(u)Δg˜j(u)-c˜ji(g)(u)δ(0)] δk˜ij(g)(u)δf˜(u1) du.
J(0,1)[f˜(u1), f˜(u2)]
=i,jl,mh˜j*(u1)h˜m(u1)k˜ij(g)*(u1)k˜lm(g)(u1)c˜il(g)(u1)×δ(u1-u2)+14 i,jl,mdu1du2 δk˜ij(g)*(u1)δf*(u1) δk˜lm(g)(u2)δf(u2)×{c˜il(g)(u1)c˜mj(g)(u1)[δ(u1-u2)]2+c˜mi(g)(u2)c˜lj(g)(u1)[δ(u1+u2)]2},
m δk˜lm(g)(u2)δf˜(u2) c˜mj(g)(u)=-mk˜lm(g)(u2) δc˜mj(g)(u)δf˜(u2).
J(0,1)[f˜(u1), f˜(u2)]
=i,jh˜i(u1)k˜ij(g)*(u1)h˜j*(u1)δ(u1-u2)-δ(0)2 i,jdu δk˜ij(g)*(u)δf*(u1) δc˜ij(g)(u)δf(u2) .
det c˜(g)(u)= |f˜(u)|4 det c˜(h)(u)+|f˜(u)|2[σ02c˜11(h)(u)+σ12c˜00(h)(u)]+σ02σ12.
δc˜ij(g)(u)δf˜(u2)=c˜ij(h)(u)[f˜*(u)δ(u-u2)+f˜(u)δ(u+u2)].
δ det c˜ij(g)(u)δf˜(u2)={2f˜*(u)|f˜(u)|2 det c˜(h)(u)+f˜*(u)[σ02c˜11(h)(u)+σ12c˜00(h)(u)]}δ(u-u2)+{2f˜(u)|f˜(u)|2 det c˜(h)(u)+f˜*(u)[σ02c˜11(h)(u)+σ12c˜00(h)(u)]}δ(u+u2).
δk˜ij(g)(u)δf˜*(u1)=(-1)i+jdet c˜(g)(u) δc˜j¯i¯(g)δf˜*(u1)-(-1)i+jc˜j¯i¯(g)[det c˜(g)(u)]2 δ det c˜(g)(u)δf˜*(u1).
J(0,1)[f˜(u1), f˜(u2)]
=i,jh˜i(u1)k˜ij(g)*(u1)h˜j*(u1)δ(u1-u2)-2|f˜(u1)|2 δ(0)δ(u1-u2)2 i,j(-1)i+jc˜ji(h)(u1)×c˜j¯i¯(h)det c˜(g)(u1)-[f˜(u1)|2c˜j¯i¯(h)+σi¯2δij][det c˜(g)(u1)]2×[2|f˜(u1)|2det c˜(h)(u1)+(σ02c˜11(h)(u1)+σ12c˜00(h)(u1)].
i,j(-1)i+jc˜ji(h)(u1)c˜j¯i¯(h)(u1)=2 det c˜(h)(u1),
J(0,1)[f˜(u1), f˜(u2)]=δ(u1-u2)i,jh˜i(u1)k˜ij(g)*(u1)×h˜j*(u1)+δ(0)|f˜(u1)|2×|α(u1)|2-2 det c˜(h)(u1)det c˜(g)(u1),
α(u)=2|f˜(u)|2det c˜(h)(u)+[σ02c˜11(h)(u)+σ12c˜00(h)(u)]det c˜(g)(u).
δk˜ij(g)(u)δf˜*(u1)
-1[c˜00(g)(u)]2 δc˜00(g)(u)δf˜*(u1).
J(0)[f˜(u1), f˜(u2)]=δ(u1-u2)|h˜0(u1)|2c˜00(g)(u1)+δ(0)|f˜(u1)|2|β0(u1)|2,
βi(u)=c˜ii(h)(u)c˜ii(g)(u).
δ(u1-u2)=limΔu0 δu1,u2Δu=δ(0)δu1,u2,
δPδf˜(u)=1Δu Pˆf^u=δ(0) Pˆf^u,
J^(0,1)[fˆ(u1), fˆ(u2)]=δu1,u2i,jh^i(u1)k^ij(g)*(u1)×h^j*(u1)+|fˆ(u1)|2|αˆ(u1)|2-2 det c^(h)(u1)det c^(g)(u1),
J^(0)[fˆ(u1), fˆ(u2)]=δu1,u2|h˜0(u1)|2c^00(g)(u1)+|fˆ(u1)|2|βˆ(u1)|2,
αˆ(u)=2|f˜(u)|2det c^(h)(u)+[σ^02c^11(h)(u)+σ^12c^00(h)(u)]det c^(g)(u),
β^i(u)=c^ii(h)(u)c^ii(g)(u).
αˆ(u)=(σ^02+σ^12)c^00(h)(u)|f˜(u)|2(σ02+σ12)c˜00(h)+σ^02σ^12=c^00(h)(u)|f˜(u)|2c˜00(h)+σ^02σ^12/(σ^02+σ^12).
J^(0,1)[fˆ(u1), fˆ(u2)]=δu1,u2|fˆ(u1)|2|αˆ(u1)|2,
αˆ(u)=σ^12c^00(h)(u)|fˆ(u)|2σ^12c^00(h)(u)+σ^02σ^12=βˆ(u).
χi(u)=|g^i(u)|2-σ^i2σ^i2c^ii(g)(u)-σ^i2σ^i2=|fˆ(u)|2c^ii(h)(u)σ^i2,
αˆ(u)=2|f˜(u)|21+121χ0(u)+1χ1(u)c^00(h)(u)c^11(h)(u)det c^(h)(u)1+1χ0(u)+1χ1(u)+1χ0(u)χ1(u)c^00(h)(u)c^11(h)(u)det c^(h)(u);
βˆ(u)=1|f˜(u)|211+1χ0(u) .
2 det c^(h)(u1)det c^(g)(u1)=2|f˜(u)|411+1χ0(u)+1χ1(u)+1χ0(u)χ1(u)c^00(h)(u)c^11(h)(u)det c^(h)(u).
hi(x)=1(λF)2Ad2ρP(ρ)×exp{i[2π/(λF)x  ρ+ϕ(ρ)+δi1τρ2]}2,
h˜i(u)=1Ad2ρP(ρ)P(ρ-uλF)exp{i[ϕ(ρ)-ϕ(ρ-uλF)]+δi1τ[ρ2-(ρ-uλF)2]},
h˜i(u)=1Ad2ρP(ρ)P(ρ-uλF)exp{iδi1τ[ρ2-(ρ-uλF)2]}h˜a(u).
h˜a(u)=exp{-(1/2)[ϕ(ρ)-ϕ(ρ-uλF)]2}=exp[-3.44(uλF/r0)5/3],
hi(x)hj(x)=1(λF)4A2d2ρP(ρ)d2rP(r)×d2ρP(ρ)d2rP(r)×exp{i2π[x(ρ-r)+x(ρ-r)]/(λF)}×exp{i[δi1τ(ρ2-r2)+δj1τ(ρ2-r2)]}×exp{i[ϕ(ρ)-ϕ(r)+ϕ(ρ)-ϕ(r)]}.
exp{i[ϕ(ρ)-ϕ(r)+ϕ(ρ)-ϕ(r)]}exp{i[ϕ(ρ)-ϕ(r)]}exp{i[ϕ(ρ)-ϕ(r)]}+exp{i[ϕ(ρ)-ϕ(r)]}exp{i[ϕ(ρ)-ϕ(r)]}.
Δhi(x)Δhj(x)=hi(x)hj(x)-hi(x)hj(x)1(λF)4A2d2ρP(ρ)d2rP(r)×d2ρP(ρ)d2rP(r)×exp{i2π[x(ρ-r)+x(ρ-r)]/(λF)}×exp{i[δi1τ(ρ2-r2)+δj1τ(ρ2-r2)]}×exp{-(1/2)[Dϕ(|ρ-r|)+Dϕ(|r-ρ|)]},
exp{i[ϕ(ρ)-ϕ(r)]}=exp[-(1/2)Dϕ(|ρ-r|)]
Δh0(x)Δh0(x)=1(λF)4A2d2ρP(ρ)exp{i2π×[(x-x)ρ]/(λF)}2d2R×exp[i2πx  R/(λF)]×exp[-(1/2)Dϕ(R)]2.
d2R exp[-(1/2)Dϕ(R)]=r026π5(3.44)6/5 Γ(6/5)=0.786r02.
Δh0(x)Δh0(x)0.618 r04(λF)2A hdiff(x-x).
Δh1(x)Δh0(x)=1(λF)4A2d2ρP(ρ)exp{i2π×[(x-x)ρ]/(λF)}exp[iτρ2]2×d2R exp[i2πx  R/(λF)]exp[-(1/2)Dϕ(R)]2.
Δh1(x)Δh0(x)0.618 r04(λF)2A hτ(x-x),
hτ(x-x)=1(λF)2Ad2ρP(ρ)exp{i[2π/(λF)(x-x)ρ+τρ2]}2.
Δh1(x)Δh1(x)1(λF)2Ad2ρP(ρ)×exp{i[2π/(λF)(x-x)ρ]}×K(ρ, τ)2,
K(ρ, τ)=1λFAd2R exp{iτ[(R+ρ)2-ρ2]}exp[-(1/2)Dϕ(R)].
kij=(-1)i+jFji(c)detC.
Fji(c)=P(-1)Pc1P1c(N-1)PN-1,
i,jcjikij=i,jcji(-1)i+jP(-1)Pc1P1c(N-1)PN-1detC.
P(-1)Pc1P1cNPN,
i,jcjikij=(detC)detC.
i,jcjikij=N.
i,jcjikij=-i,jcjikij.
(lndet C)=-i,jcjikij.
P(x1, x2,, xn)=P(x1)P(x2|x1)×P(x3|x2, x1)P(xn|xn-1,, x1).
ln P(x1, x2,, xn)=i=1nln P(xi|xi-1,, x1).
 ln P(xj|xj-1,, x1)θ  ln P(xi|xi-1,, x1)θ= ln P(xj|xj-1,, x1)θ ln P(xi|xi-1,, x1)θ×P(xi|xi-1,, x1)P(xi-1,, x1)m=1idxm= ln P(xj|xj-1,, x1)θθP(xi|xi-1,, x1)dxi
×P(xi-1,, x1)m=1i-1dxm,
J[x1, x2,, xn]=i=1nJ[xi|xi-1,, x1].

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