Abstract

Maximum-likelihood estimation techniques are presented for the problem of forming object estimates from turbulence-degraded images when the point-spread functions are unknown. The inability of unconstrained maximum-likelihood methods to form meaningful estimates is acknowledged, and iterative algorithms are derived for estimating the object by using both a penalized maximum-likelihood method and a physically meaningful parameterization of the point-spread functions by phase errors distributed over an aperture.

© 1993 Optical Society of America

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  1. A. Labeyrie, “Attainment of diffraction-limited resolution in large telescopes by Fourier analysing speckle patterns in star images,” Astron. Astrophys. 6, 85–87 (1970).
  2. K. T. Knox, B. J. Thompson, “Recovery of images from atmospherically degraded short-exposure photographs,” Astrophys. J. Lett. 193, L45–L48 (1974).
    [CrossRef]
  3. A. W. Lohmann, G. Weigelt, B. Wirnitzer, “Speckle masking in astronomy: triple correlation theory and applications,” Appl. Opt. 22, 4028–4037 (1983).
    [CrossRef] [PubMed]
  4. H. Bartelt, A. W. Lohmann, B. Wirnitzer, “Phase and amplitude recovery from bispectra,” Appl. Opt. 23, 3121–3129 (1984).
    [CrossRef] [PubMed]
  5. C. Dainty, J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery: Theory and Applications, H. Stark, ed. (Academic, New York, 1987), Chap. 7, pp. 231–275.
  6. J. Primot, G. Rousset, J. Fontanella, “Deconvolution from wavefront sensing: a new technique for compensating turbulence-degraded images,” J. Opt. Soc. Am. A 7, 1598–1608 (1990).
    [CrossRef]
  7. J. D. Gonglewski, D. G. Voelz, J. S. Fender, D. C. Dayton, B. K. Spielbusch, R. E. Pierson, “First astronomical application of postdetection turbulence compensation: images of αAurigae, νUrsae Majoris, and αGerminorum using self-referenced speckle holography,” Appl. Opt. 29, 4527–4529 (1990).
    [CrossRef] [PubMed]
  8. J. H. Hardy, “Active optics: a new technology for the control of light,” Proc. IEEE 66, 651–697 (1978).
    [CrossRef]
  9. C. S. Gardner, B. M. Welsh, L. A. Thompson, “Design and performance analysis of adaptive optical telescopes using laser guide stars,” Proc. IEEE 78, 1721–1743 (1990).
    [CrossRef]
  10. P. Nisenson, R. Barakat, “Partial atmospheric correction with adaptive optics,” J. Opt. Soc. Am. A 4, 2249–2253 (1987).
    [CrossRef]
  11. M. C. Roggemann, “Limited degree-of-freedom adaptive optics and image reconstruction,” Appl. Opt. 30, 4227–4233 (1991).
    [CrossRef] [PubMed]
  12. T. G. Stockham, T. M. Cannon, R. B. Ingebretsen, “Blind deconvolution through digital signal processing,” Proc. IEEE 63, 678–692 (1975).
    [CrossRef]
  13. R. G. Lane, R. H. T. Bates, “Automatic multidimensional deconvolution,” J. Opt. Soc. Am. A 4, 180–188 (1987).
    [CrossRef]
  14. R. L. Lagendijk, J. Biemond, D. E. Boekee, “Identification and restoration of noisy blurred images using the expectation-maximization algorithm,”IEEE Trans. Acoust. Speech Signal Process 38, 1180–1191 (1990).
    [CrossRef]
  15. G. R. Ayers, J. C. Dainty, “Iterative blind deconvolution method and its application,” Opt. Lett. 13, 547–549 (1988).
    [CrossRef] [PubMed]
  16. T. J. Holmes, “Blind deconvolution of quantum-limited incoherent imagery: maximum-likelihood approach,” J. Opt. Soc. Am. A 9, 1052–1061 (1992).
    [CrossRef] [PubMed]
  17. N. Miura, N. Baba, “Extended-object reconstruction with sequential use of the iterative blind deconvolution method,” Opt. Commun. 89, 375–379 (1992).
    [CrossRef]
  18. D. L. Snyder, M. I. Miller, Random Point Processes in Time and Space (Springer-Verlag, New York, 1991).
    [CrossRef]
  19. J. R. Thompson, R. A. Tapia, Nonparametric Function Estimation, Modeling, and Simulation (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1990).
    [CrossRef]
  20. A. P. Dempster, N. M. Laird, D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,”J. R. Stat. Soc. B 39, 1–37 (1977).
  21. C. R. Jeff Wu, “On the convergence properties of the EM algorithm,” Ann. Stat. 11, 95–103 (1983).
    [CrossRef]
  22. L. A. Shepp, Y. Vardi, “Maximum-likelihood reconstruction for emission tomography,”IEEE Trans. Med. Imag. MI-1, 113–121 (1982).
    [CrossRef]
  23. D. G. Luenberger, Optimization by Vector Space Methods (Wiley, New York, 1969).
  24. R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).
  25. J. R. Fienup, “Reconstruction and synthesis applications of an iterative algorithm,” in Transformations in Optical Signal Processing, W. T. Rhodes, ed., Proc. Soc. Photo-Opt. Instrum. Eng.373, 147–160 (1981).
    [CrossRef]

1992 (2)

N. Miura, N. Baba, “Extended-object reconstruction with sequential use of the iterative blind deconvolution method,” Opt. Commun. 89, 375–379 (1992).
[CrossRef]

T. J. Holmes, “Blind deconvolution of quantum-limited incoherent imagery: maximum-likelihood approach,” J. Opt. Soc. Am. A 9, 1052–1061 (1992).
[CrossRef] [PubMed]

1991 (1)

1990 (4)

J. Primot, G. Rousset, J. Fontanella, “Deconvolution from wavefront sensing: a new technique for compensating turbulence-degraded images,” J. Opt. Soc. Am. A 7, 1598–1608 (1990).
[CrossRef]

J. D. Gonglewski, D. G. Voelz, J. S. Fender, D. C. Dayton, B. K. Spielbusch, R. E. Pierson, “First astronomical application of postdetection turbulence compensation: images of αAurigae, νUrsae Majoris, and αGerminorum using self-referenced speckle holography,” Appl. Opt. 29, 4527–4529 (1990).
[CrossRef] [PubMed]

R. L. Lagendijk, J. Biemond, D. E. Boekee, “Identification and restoration of noisy blurred images using the expectation-maximization algorithm,”IEEE Trans. Acoust. Speech Signal Process 38, 1180–1191 (1990).
[CrossRef]

C. S. Gardner, B. M. Welsh, L. A. Thompson, “Design and performance analysis of adaptive optical telescopes using laser guide stars,” Proc. IEEE 78, 1721–1743 (1990).
[CrossRef]

1988 (1)

1987 (2)

1984 (1)

1983 (2)

1982 (1)

L. A. Shepp, Y. Vardi, “Maximum-likelihood reconstruction for emission tomography,”IEEE Trans. Med. Imag. MI-1, 113–121 (1982).
[CrossRef]

1978 (1)

J. H. Hardy, “Active optics: a new technology for the control of light,” Proc. IEEE 66, 651–697 (1978).
[CrossRef]

1977 (1)

A. P. Dempster, N. M. Laird, D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,”J. R. Stat. Soc. B 39, 1–37 (1977).

1975 (1)

T. G. Stockham, T. M. Cannon, R. B. Ingebretsen, “Blind deconvolution through digital signal processing,” Proc. IEEE 63, 678–692 (1975).
[CrossRef]

1974 (1)

K. T. Knox, B. J. Thompson, “Recovery of images from atmospherically degraded short-exposure photographs,” Astrophys. J. Lett. 193, L45–L48 (1974).
[CrossRef]

1972 (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

1970 (1)

A. Labeyrie, “Attainment of diffraction-limited resolution in large telescopes by Fourier analysing speckle patterns in star images,” Astron. Astrophys. 6, 85–87 (1970).

Ayers, G. R.

Baba, N.

N. Miura, N. Baba, “Extended-object reconstruction with sequential use of the iterative blind deconvolution method,” Opt. Commun. 89, 375–379 (1992).
[CrossRef]

Barakat, R.

Bartelt, H.

Bates, R. H. T.

Biemond, J.

R. L. Lagendijk, J. Biemond, D. E. Boekee, “Identification and restoration of noisy blurred images using the expectation-maximization algorithm,”IEEE Trans. Acoust. Speech Signal Process 38, 1180–1191 (1990).
[CrossRef]

Boekee, D. E.

R. L. Lagendijk, J. Biemond, D. E. Boekee, “Identification and restoration of noisy blurred images using the expectation-maximization algorithm,”IEEE Trans. Acoust. Speech Signal Process 38, 1180–1191 (1990).
[CrossRef]

Cannon, T. M.

T. G. Stockham, T. M. Cannon, R. B. Ingebretsen, “Blind deconvolution through digital signal processing,” Proc. IEEE 63, 678–692 (1975).
[CrossRef]

Dainty, C.

C. Dainty, J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery: Theory and Applications, H. Stark, ed. (Academic, New York, 1987), Chap. 7, pp. 231–275.

Dainty, J. C.

Dayton, D. C.

Dempster, A. P.

A. P. Dempster, N. M. Laird, D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,”J. R. Stat. Soc. B 39, 1–37 (1977).

Fender, J. S.

Fienup, J. R.

C. Dainty, J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery: Theory and Applications, H. Stark, ed. (Academic, New York, 1987), Chap. 7, pp. 231–275.

J. R. Fienup, “Reconstruction and synthesis applications of an iterative algorithm,” in Transformations in Optical Signal Processing, W. T. Rhodes, ed., Proc. Soc. Photo-Opt. Instrum. Eng.373, 147–160 (1981).
[CrossRef]

Fontanella, J.

Gardner, C. S.

C. S. Gardner, B. M. Welsh, L. A. Thompson, “Design and performance analysis of adaptive optical telescopes using laser guide stars,” Proc. IEEE 78, 1721–1743 (1990).
[CrossRef]

Gerchberg, R. W.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Gonglewski, J. D.

Hardy, J. H.

J. H. Hardy, “Active optics: a new technology for the control of light,” Proc. IEEE 66, 651–697 (1978).
[CrossRef]

Holmes, T. J.

Ingebretsen, R. B.

T. G. Stockham, T. M. Cannon, R. B. Ingebretsen, “Blind deconvolution through digital signal processing,” Proc. IEEE 63, 678–692 (1975).
[CrossRef]

Jeff Wu, C. R.

C. R. Jeff Wu, “On the convergence properties of the EM algorithm,” Ann. Stat. 11, 95–103 (1983).
[CrossRef]

Knox, K. T.

K. T. Knox, B. J. Thompson, “Recovery of images from atmospherically degraded short-exposure photographs,” Astrophys. J. Lett. 193, L45–L48 (1974).
[CrossRef]

Labeyrie, A.

A. Labeyrie, “Attainment of diffraction-limited resolution in large telescopes by Fourier analysing speckle patterns in star images,” Astron. Astrophys. 6, 85–87 (1970).

Lagendijk, R. L.

R. L. Lagendijk, J. Biemond, D. E. Boekee, “Identification and restoration of noisy blurred images using the expectation-maximization algorithm,”IEEE Trans. Acoust. Speech Signal Process 38, 1180–1191 (1990).
[CrossRef]

Laird, N. M.

A. P. Dempster, N. M. Laird, D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,”J. R. Stat. Soc. B 39, 1–37 (1977).

Lane, R. G.

Lohmann, A. W.

Luenberger, D. G.

D. G. Luenberger, Optimization by Vector Space Methods (Wiley, New York, 1969).

Miller, M. I.

D. L. Snyder, M. I. Miller, Random Point Processes in Time and Space (Springer-Verlag, New York, 1991).
[CrossRef]

Miura, N.

N. Miura, N. Baba, “Extended-object reconstruction with sequential use of the iterative blind deconvolution method,” Opt. Commun. 89, 375–379 (1992).
[CrossRef]

Nisenson, P.

Pierson, R. E.

Primot, J.

Roggemann, M. C.

Rousset, G.

Rubin, D. B.

A. P. Dempster, N. M. Laird, D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,”J. R. Stat. Soc. B 39, 1–37 (1977).

Saxton, W. O.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Shepp, L. A.

L. A. Shepp, Y. Vardi, “Maximum-likelihood reconstruction for emission tomography,”IEEE Trans. Med. Imag. MI-1, 113–121 (1982).
[CrossRef]

Snyder, D. L.

D. L. Snyder, M. I. Miller, Random Point Processes in Time and Space (Springer-Verlag, New York, 1991).
[CrossRef]

Spielbusch, B. K.

Stockham, T. G.

T. G. Stockham, T. M. Cannon, R. B. Ingebretsen, “Blind deconvolution through digital signal processing,” Proc. IEEE 63, 678–692 (1975).
[CrossRef]

Tapia, R. A.

J. R. Thompson, R. A. Tapia, Nonparametric Function Estimation, Modeling, and Simulation (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1990).
[CrossRef]

Thompson, B. J.

K. T. Knox, B. J. Thompson, “Recovery of images from atmospherically degraded short-exposure photographs,” Astrophys. J. Lett. 193, L45–L48 (1974).
[CrossRef]

Thompson, J. R.

J. R. Thompson, R. A. Tapia, Nonparametric Function Estimation, Modeling, and Simulation (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1990).
[CrossRef]

Thompson, L. A.

C. S. Gardner, B. M. Welsh, L. A. Thompson, “Design and performance analysis of adaptive optical telescopes using laser guide stars,” Proc. IEEE 78, 1721–1743 (1990).
[CrossRef]

Vardi, Y.

L. A. Shepp, Y. Vardi, “Maximum-likelihood reconstruction for emission tomography,”IEEE Trans. Med. Imag. MI-1, 113–121 (1982).
[CrossRef]

Voelz, D. G.

Weigelt, G.

Welsh, B. M.

C. S. Gardner, B. M. Welsh, L. A. Thompson, “Design and performance analysis of adaptive optical telescopes using laser guide stars,” Proc. IEEE 78, 1721–1743 (1990).
[CrossRef]

Wirnitzer, B.

Ann. Stat. (1)

C. R. Jeff Wu, “On the convergence properties of the EM algorithm,” Ann. Stat. 11, 95–103 (1983).
[CrossRef]

Appl. Opt. (4)

Astron. Astrophys. (1)

A. Labeyrie, “Attainment of diffraction-limited resolution in large telescopes by Fourier analysing speckle patterns in star images,” Astron. Astrophys. 6, 85–87 (1970).

Astrophys. J. Lett. (1)

K. T. Knox, B. J. Thompson, “Recovery of images from atmospherically degraded short-exposure photographs,” Astrophys. J. Lett. 193, L45–L48 (1974).
[CrossRef]

IEEE Trans. Acoust. Speech Signal Process (1)

R. L. Lagendijk, J. Biemond, D. E. Boekee, “Identification and restoration of noisy blurred images using the expectation-maximization algorithm,”IEEE Trans. Acoust. Speech Signal Process 38, 1180–1191 (1990).
[CrossRef]

IEEE Trans. Med. Imag. (1)

L. A. Shepp, Y. Vardi, “Maximum-likelihood reconstruction for emission tomography,”IEEE Trans. Med. Imag. MI-1, 113–121 (1982).
[CrossRef]

J. Opt. Soc. Am. A (4)

J. R. Stat. Soc. B (1)

A. P. Dempster, N. M. Laird, D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,”J. R. Stat. Soc. B 39, 1–37 (1977).

Opt. Commun. (1)

N. Miura, N. Baba, “Extended-object reconstruction with sequential use of the iterative blind deconvolution method,” Opt. Commun. 89, 375–379 (1992).
[CrossRef]

Opt. Lett. (1)

Optik (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Proc. IEEE (3)

T. G. Stockham, T. M. Cannon, R. B. Ingebretsen, “Blind deconvolution through digital signal processing,” Proc. IEEE 63, 678–692 (1975).
[CrossRef]

J. H. Hardy, “Active optics: a new technology for the control of light,” Proc. IEEE 66, 651–697 (1978).
[CrossRef]

C. S. Gardner, B. M. Welsh, L. A. Thompson, “Design and performance analysis of adaptive optical telescopes using laser guide stars,” Proc. IEEE 78, 1721–1743 (1990).
[CrossRef]

Other (5)

C. Dainty, J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery: Theory and Applications, H. Stark, ed. (Academic, New York, 1987), Chap. 7, pp. 231–275.

D. L. Snyder, M. I. Miller, Random Point Processes in Time and Space (Springer-Verlag, New York, 1991).
[CrossRef]

J. R. Thompson, R. A. Tapia, Nonparametric Function Estimation, Modeling, and Simulation (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1990).
[CrossRef]

J. R. Fienup, “Reconstruction and synthesis applications of an iterative algorithm,” in Transformations in Optical Signal Processing, W. T. Rhodes, ed., Proc. Soc. Photo-Opt. Instrum. Eng.373, 147–160 (1981).
[CrossRef]

D. G. Luenberger, Optimization by Vector Space Methods (Wiley, New York, 1969).

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Figures (3)

Fig. 1
Fig. 1

Results of a computer-generated example produced by using the penalized maximum-likelihood technique. The first row contains the unknown object intensity. The second row contains the two unknown point-spread functions. The third row contains the two detected images (300,000 photons in each image). The fourth row contains the estimated point-spread functions. The fifth row contains the object estimate.

Fig. 2
Fig. 2

Results of the application of the penalized maximum-likelihood technique to real data. The first row contains two of the eight measured specklegrams. The second row contains the estimates of the point-spread functions. The third row contains the object estimate, magnified by a factor of 4.

Fig. 3
Fig. 3

Results of a computer-generated example produced by using the parameterized, phase-error model. The first row contains the unknown object intensity. The second row contains the two unknown point-spread functions. The third row contains the two detected images (50,000 photons in each image). The fourth row contains the estimated point-spread functions. The fifth row contains the object estimate.

Equations (53)

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i k ( y ; o , h k ) = x X h k ( y x ) o ( x ) ,             y Y
L ( o ^ , h ^ ) L ( o , h )
L ( o , h ) = k = 1 K [ - y Y i k ( y ; o , h k ) + y Y d k ( y ) ln i k ( y ; o , h k ) ] + A . T . ,
i k ( y ; α o ^ , α - 1 h ^ k ) = i k ( y ; o ^ , h ^ k ) .
A = def [ o : x o ( x ) = 1 ,             o 0 ] ,
L φ ( o , h ) = L ( o , h ) - β φ ( o ) ,
φ ( o ) = - x ln [ 1 - o ( x ) ] .
L φ ( o ^ , h ^ ) L φ ( o , h )
L φ ( o new , h new ) L φ ( o old , h old ) .
h k new ( x ) = h k old ( x ) y o old ( y - x ) i k ( y ; o old , h k old ) d k ( y ) ,
Q o ( o new o old , h old ) Q o ( o old o old , h old ) ,
Q o ( o o old , h old ) = x μ ( x ; o old , h old ) ln o ( x ) + β X ln [ 1 - o ( x ) ]
μ ( x ; o old , h old ) = o old ( x ) k y h k old ( y - x ) i k ( y ; o old , h k old ) d k ( y ) ,
L φ ( o new , h new ) L φ ( o old , h old ) .
d k ( y ) = x d ˜ k ( y x ) ,
E [ d ˜ k ( y x ) ] = h k ( y - x ) o ( x )
E [ d k ( y ) ] = x E [ d ˜ k ( y x ) ] = x h k ( y - x ) o ( x ) = i k ( y ; o , h k ) ,
L CD ( o , h ) = - k y x h k ( y - x ) o ( x ) + k y x d ˜ k ( y x ) ln h k ( y - x ) o ( x ) + A . T . ,
L φ CD ( o , h ) = L CD ( o , h ) - β φ ( o ) .
Q ( o , h o old , h old ) = def E old [ L φ CD ( o , h ) { d k ( y ) } ] ,
Q ( o new , h new o old , h old ) Q ( o old , h old o old , h old ) ,
L φ ( o new , h new ) L φ ( o old , h old ) ,
Q ( o , h o old , h old ) = - k x h k ( x ) + k y x E old [ d ˜ k ( y x ) d k ( y ) ] ln h k ( y - x ) o ( x ) + β x ln [ 1 - o ( x ) ] ,
Q ( o , h o old , h old ) = Q h ( h o old , h old ) + Q o ( o o old , h old ) ,
Q h ( h o old , h old ) = - k x h k ( x ) + k y x E old [ d ˜ k ( y x ) d k ( y ) ] ln h k ( y - x ) ,
Q h ( o o old , h old ) = x [ k y E old [ d ˜ k ( y x ) d k ( y ) ] ] ln o ( x ) + β x ln [ 1 - o ( x ) ] .
E old [ d ˜ k ( y x ) d k ( y ) ] = h k old ( y - x ) o old ( x ) i k ( y ; o old , h old ) d k ( y ) .
h k new ( x ) = y E old [ d ˜ k ( y y - x ) d k ( y ) ] = h k old ( x ) y o old ( y - x ) i k ( y ; o old , h old ) d k ( y )
k y E old [ d ˜ k ( y x ) d k ( y ) ] = μ ( x ; o old , h old ) .
Q o ( o new o old , h old ) Q o ( o old o old , h old ) .
o new = arg max o A Q o ( o o old , h old )
o new = arg max 0 o 1 [ Q o ( o o old , h old ) + γ x o ( x ) ] ,
o ˜ = arg max 0 o 1 [ Q o ( o o old , h old ) - D x o ( x ) ] ,
o new ( x ) = { o ˜ ( x ) / z o ˜ ( z ) Q 0 [ o ˜ / z o ˜ ( z ) o old , h old ] > Q o ( o old o old , h old ) o old ( x ) else .
o ˜ ( x ) = [ μ ( x ) + D + β ] - { [ μ ( x ) + D + β ] 2 - 4 D μ ( x ) } 1 / 2 2 D .
h k ( y x ; α k , θ k ) = α k | u U A ( u ) exp [ i θ k ( u ) ] exp [ - i 2 π K u ( y - x ) ] | 2 = α k g ( y - x ; θ k ) ,
g ( x ; θ k ) = def | u U A ( u ) exp [ i θ k ( u ) ] exp ( - i 2 π K u x ) | 2 ,
L ( o ^ , h ^ ) L ( o , h )
α ^ k = D k / G ,
L ( o , h ) = - k y x α k g ( y - x ; θ k ) o ( x ) + k y d k ( y ) ln α k + k y d k ( y ) ln [ x g ( y - x ; θ k ) o ( x ) ] = - G k α k + k y d k ( y ) ln α k + k y d k ( y ) ln [ x g ( y - x ; θ k ) o ( x ) ] ,
α k = D k / G ,
o new ( x ) = D - 1 o old ( x ) k y α ^ k g ( y - x ; θ k old ) i k ( y ; o old , h k old ) d k ( y ) ,
x ξ ( x ; θ k old ) ln g ( x ; θ k new ) x ξ ( x ; θ k old ) ln g ( x ; θ k old ) ,
ξ ( x ; θ k old ) = g ( x ; θ k old ) [ y o old ( y - x ) i k ( y ; o old , h k old ) d k ( y ) ]
L ( o new , h new ) L ( o old , h old ) .
x ξ ( x ; θ k old ) ln g ( x ; θ k new ) - x ξ ( x ; θ k old ) ln g ( x ; θ k old ) = x ξ ( x ; θ k old ) ln g ( x ; θ k new ) ( x ; θ k old ) = 2 x ξ ( x ; θ k old ) ln [ g ( x ; θ k new ) g ( x ; θ k old ) ] 1 / 2 2 x ξ ( x ; θ k old ) { 1 - [ g ( x ; θ k old ) g ( x ; θ k new ) ] 1 / 2 } 0 ,
G ( x ; θ k ) = | u U A ( u ) exp [ i θ k ( u ) ] exp ( - i 2 π K u x ) | 2 ,
θ k new ( u ) = phase ( u - 1 { ξ ( x ; θ k old ) × exp [ i phase { g ˜ ( x ; θ k old ) } ] } ) ,
g ˜ ( x ; θ k old ) = x { A ( u ) exp [ i θ k old ( u ) ] } ,
x [ F ( u ) ] = u F ( u ) exp ( - i 2 π K u x ) ,
x [ ξ ( x ; θ k old ) ] 1 / 2 - [ g ( x ; θ k new ) ] 1 / 2 2 - x [ ξ ( x ; θ k old ) ] 1 / 2 - [ g ( x ; θ k new ) ] 1 / 2 2 = 2 x [ ξ ( x ; θ k old ) ] 1 / 2 { [ g ( x ; θ k new ) ] 1 / 2 - [ g ( x ; θ k old ) ] 1 / 2 } = 2 x [ ξ ( x ; θ k old ) g ( x ; θ k new ) ] 1 / 2 { 1 - [ g ( x ; θ k old ) g ( x ; θ k new ) ] 1 / 2 } 0.
θ ˜ ( u ) = phase ( u - 1 { ξ ( x ; θ k old ) exp [ i phase { g ˜ ( x ; θ k old ) } ] } )
θ k new ( u ) = { θ ˜ ( u ) x ξ ( x ; θ k old ) ln g ( x ; θ ˜ ) x ξ ( x ; θ k old ) ln g ( x ; θ k old ) θ k old ( u ) else .

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