Abstract

The joint estimation of an object and the aberrations of an incoherent imaging system from multiple images incorporating phase diversity is investigated. Maximum-likelihood estimation is considered under additive Gaussian and Poisson noise models. Expressions for an aberration-only objective function that accommodates an arbitrary number of diversity images and its gradient are derived for the case of a Gaussian noise model. Expressions for the log-likelihood function and its gradient are presented for the case of Poisson noise. An expectation-maximization algorithm that enforces a nonnegativity constraint in a natural fashion is constructed for use in the Poisson noise case.

© 1992 Optical Society of America

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  1. J. D. Gonglewski, C. R. DeHainaut, C. M. Lampkin, R. C. Dymale, “System design of a wavefront sensing package for a wide field of view optical phased array,” Opt. Eng. 27, 785–792 (1988).
    [CrossRef]
  2. J. N. Cederquist, J. R. Fienup, C. C. Wackerman, S. R. Robinson, D. Kryskowski, “Wave-front phase estimation from Fourier intensity measurements,” J. Opt. Soc. Am. A 6, 1020–1026 (1989).
    [CrossRef]
  3. R. G. Lyon, “HST phase retrieval: a parameter estimation,” in Applications of Digital Image Processing XIV, A. G. Tescher, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1567, 317–326 (1991).
    [CrossRef]
  4. J. R. Fienup, “Phase retrieval for the Hubble Space Telescope using iterative propagation algorithms,” in Applications of Digital Image Processing XIV, A. G. Tescher, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1567, 327–332 (1991).
    [CrossRef]
  5. R. A. Gonsalves, R. Childlaw, “Wavefront sensing by phase retrieval,” in Applications of Digital Image Processing III, A. G. Tescher, ed., Proc. Soc. Photo-Opt. Instrum. Eng.207, 32–39 (1979).
    [CrossRef]
  6. R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21, 829–832 (1982).
    [CrossRef]
  7. R. G. Paxman, J. R. Fienup, “Optical misalignment sensing and image reconstruction using phase diversity,” J. Opt. Soc. Am. A 5, 914–923 (1988).
    [CrossRef]
  8. R. G. Paxman, S. L. Crippen, “Aberration correction for phased-array telescopes using phase diversity,” in Digital Image Synthesis and Inverse Optics, A. F. Gmitro, P. S. Idell, I. J. LaHaie, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1351, 787–797 (1990).
    [CrossRef]
  9. J. M. Mendel, Lessons in Digital Estimation Theory (Prentice-Hall, Englewood Cliffs, N.J., 1987).
  10. A. V. Oppenheim, R. W. Schafer, Discrete-Time Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1989).
  11. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968).
  12. F. Roddier, “Passive versus active methods in optical interferometry,” in Proceedings of the ESO/NOAO Conference on High Resolution Imaging by Interferometry, F. Merkle, ed. (European Southern Observatory, Garching, Federal Republic of Germany, 1988), pp. 565–574.
  13. J. Primot, G. Rousset, J. C. Fontanella, “Deconvolution from wave-front sensing: a new technique for compensating turbulence-degraded images,” J. Opt. Soc. Am. A 7, 1598–1608 (1990).
    [CrossRef]
  14. 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 αGeminorum using self-referenced speckle holography,” Appl. Opt. 29, 4527–4529 (1990).
    [CrossRef] [PubMed]
  15. D. C. Luenberger, Linear and Nonlinear Programming (Addison-Wesley, Reading, Mass., 1984).
  16. D. L. Snyder, D. G. Politte, “Image reconstruction from list-mode data in an emission tomography system having time-of-flight measurements,”IEEE Trans. Nucl. Sci. NS-30, 1843–1849 (1983).
    [CrossRef]
  17. D. G. Politte, “Reconstruction algorithms for time-of-flight assisted positron-emission tomographs,” M.S. thesis (Sever Institute of Technology, Washington University, St. Louis, Mo., 1983).
  18. L. Kaufman, “Implementing and accelerating the EM algorithm for positron emission tomography,”IEEE Trans. Med. Imag. MI-6, 37–51 (1987).
    [CrossRef]
  19. A. P. Dempster, N. M. Laird, D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,”J. R. Stat. Soc. Ser. B 39, 1–38 (1977).
  20. C. F. J. Wu, “On the convergence properties of the EM algorithm,” Ann. Stat. 11, 95–103 (1983).
    [CrossRef]
  21. Y. Vardi, L. A. Shepp, L. Kaufman, “A statistical model for positron emission tomography,”J. Am. Stat. Assoc. 80, 8–38 (1985).
    [CrossRef]
  22. W. H. Richardson, “Bayesian-based iterative method of image restoration,”J. Opt. Soc. Am. 62, 55–59 (1972).
    [CrossRef]
  23. L. B. Lucy, “An iterative technique for the rectification of observed distributions,” Astron. J. 79, 745–754 (1974).
    [CrossRef]
  24. N. B. Baranova, B. Y. Zel’dovich, “Dislocations of the wave-front surface and zeros of the amplitude,” Sov. Phys. JETP 53, 925–929 (1981).
  25. L. A. Shepp, Y. Vardi, “Maximum-likelihood reconstruction for emission tomography,”IEEE Trans. Med. Imag. MI-1, 113–121 (1982).
    [CrossRef]

1990 (2)

1989 (1)

1988 (2)

J. D. Gonglewski, C. R. DeHainaut, C. M. Lampkin, R. C. Dymale, “System design of a wavefront sensing package for a wide field of view optical phased array,” Opt. Eng. 27, 785–792 (1988).
[CrossRef]

R. G. Paxman, J. R. Fienup, “Optical misalignment sensing and image reconstruction using phase diversity,” J. Opt. Soc. Am. A 5, 914–923 (1988).
[CrossRef]

1987 (1)

L. Kaufman, “Implementing and accelerating the EM algorithm for positron emission tomography,”IEEE Trans. Med. Imag. MI-6, 37–51 (1987).
[CrossRef]

1985 (1)

Y. Vardi, L. A. Shepp, L. Kaufman, “A statistical model for positron emission tomography,”J. Am. Stat. Assoc. 80, 8–38 (1985).
[CrossRef]

1983 (2)

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

D. L. Snyder, D. G. Politte, “Image reconstruction from list-mode data in an emission tomography system having time-of-flight measurements,”IEEE Trans. Nucl. Sci. NS-30, 1843–1849 (1983).
[CrossRef]

1982 (2)

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

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

1981 (1)

N. B. Baranova, B. Y. Zel’dovich, “Dislocations of the wave-front surface and zeros of the amplitude,” Sov. Phys. JETP 53, 925–929 (1981).

1977 (1)

A. P. Dempster, N. M. Laird, D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,”J. R. Stat. Soc. Ser. B 39, 1–38 (1977).

1974 (1)

L. B. Lucy, “An iterative technique for the rectification of observed distributions,” Astron. J. 79, 745–754 (1974).
[CrossRef]

1972 (1)

Baranova, N. B.

N. B. Baranova, B. Y. Zel’dovich, “Dislocations of the wave-front surface and zeros of the amplitude,” Sov. Phys. JETP 53, 925–929 (1981).

Cederquist, J. N.

Childlaw, R.

R. A. Gonsalves, R. Childlaw, “Wavefront sensing by phase retrieval,” in Applications of Digital Image Processing III, A. G. Tescher, ed., Proc. Soc. Photo-Opt. Instrum. Eng.207, 32–39 (1979).
[CrossRef]

Crippen, S. L.

R. G. Paxman, S. L. Crippen, “Aberration correction for phased-array telescopes using phase diversity,” in Digital Image Synthesis and Inverse Optics, A. F. Gmitro, P. S. Idell, I. J. LaHaie, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1351, 787–797 (1990).
[CrossRef]

Dayton, D. C.

DeHainaut, C. R.

J. D. Gonglewski, C. R. DeHainaut, C. M. Lampkin, R. C. Dymale, “System design of a wavefront sensing package for a wide field of view optical phased array,” Opt. Eng. 27, 785–792 (1988).
[CrossRef]

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. Ser. B 39, 1–38 (1977).

Dymale, R. C.

J. D. Gonglewski, C. R. DeHainaut, C. M. Lampkin, R. C. Dymale, “System design of a wavefront sensing package for a wide field of view optical phased array,” Opt. Eng. 27, 785–792 (1988).
[CrossRef]

Fender, J. S.

Fienup, J. R.

Fontanella, J. C.

Gonglewski, J. D.

Gonsalves, R. A.

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

R. A. Gonsalves, R. Childlaw, “Wavefront sensing by phase retrieval,” in Applications of Digital Image Processing III, A. G. Tescher, ed., Proc. Soc. Photo-Opt. Instrum. Eng.207, 32–39 (1979).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968).

Kaufman, L.

L. Kaufman, “Implementing and accelerating the EM algorithm for positron emission tomography,”IEEE Trans. Med. Imag. MI-6, 37–51 (1987).
[CrossRef]

Y. Vardi, L. A. Shepp, L. Kaufman, “A statistical model for positron emission tomography,”J. Am. Stat. Assoc. 80, 8–38 (1985).
[CrossRef]

Kryskowski, D.

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. Ser. B 39, 1–38 (1977).

Lampkin, C. M.

J. D. Gonglewski, C. R. DeHainaut, C. M. Lampkin, R. C. Dymale, “System design of a wavefront sensing package for a wide field of view optical phased array,” Opt. Eng. 27, 785–792 (1988).
[CrossRef]

Lucy, L. B.

L. B. Lucy, “An iterative technique for the rectification of observed distributions,” Astron. J. 79, 745–754 (1974).
[CrossRef]

Luenberger, D. C.

D. C. Luenberger, Linear and Nonlinear Programming (Addison-Wesley, Reading, Mass., 1984).

Lyon, R. G.

R. G. Lyon, “HST phase retrieval: a parameter estimation,” in Applications of Digital Image Processing XIV, A. G. Tescher, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1567, 317–326 (1991).
[CrossRef]

Mendel, J. M.

J. M. Mendel, Lessons in Digital Estimation Theory (Prentice-Hall, Englewood Cliffs, N.J., 1987).

Oppenheim, A. V.

A. V. Oppenheim, R. W. Schafer, Discrete-Time Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1989).

Paxman, R. G.

R. G. Paxman, J. R. Fienup, “Optical misalignment sensing and image reconstruction using phase diversity,” J. Opt. Soc. Am. A 5, 914–923 (1988).
[CrossRef]

R. G. Paxman, S. L. Crippen, “Aberration correction for phased-array telescopes using phase diversity,” in Digital Image Synthesis and Inverse Optics, A. F. Gmitro, P. S. Idell, I. J. LaHaie, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1351, 787–797 (1990).
[CrossRef]

Pierson, R. E.

Politte, D. G.

D. L. Snyder, D. G. Politte, “Image reconstruction from list-mode data in an emission tomography system having time-of-flight measurements,”IEEE Trans. Nucl. Sci. NS-30, 1843–1849 (1983).
[CrossRef]

D. G. Politte, “Reconstruction algorithms for time-of-flight assisted positron-emission tomographs,” M.S. thesis (Sever Institute of Technology, Washington University, St. Louis, Mo., 1983).

Primot, J.

Richardson, W. H.

Robinson, S. R.

Roddier, F.

F. Roddier, “Passive versus active methods in optical interferometry,” in Proceedings of the ESO/NOAO Conference on High Resolution Imaging by Interferometry, F. Merkle, ed. (European Southern Observatory, Garching, Federal Republic of Germany, 1988), pp. 565–574.

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. Ser. B 39, 1–38 (1977).

Schafer, R. W.

A. V. Oppenheim, R. W. Schafer, Discrete-Time Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1989).

Shepp, L. A.

Y. Vardi, L. A. Shepp, L. Kaufman, “A statistical model for positron emission tomography,”J. Am. Stat. Assoc. 80, 8–38 (1985).
[CrossRef]

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, D. G. Politte, “Image reconstruction from list-mode data in an emission tomography system having time-of-flight measurements,”IEEE Trans. Nucl. Sci. NS-30, 1843–1849 (1983).
[CrossRef]

Spielbusch, B. K.

Vardi, Y.

Y. Vardi, L. A. Shepp, L. Kaufman, “A statistical model for positron emission tomography,”J. Am. Stat. Assoc. 80, 8–38 (1985).
[CrossRef]

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

Voelz, D. G.

Wackerman, C. C.

Wu, C. F. J.

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

Zel’dovich, B. Y.

N. B. Baranova, B. Y. Zel’dovich, “Dislocations of the wave-front surface and zeros of the amplitude,” Sov. Phys. JETP 53, 925–929 (1981).

Ann. Stat. (1)

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

Appl. Opt. (1)

Astron. J. (1)

L. B. Lucy, “An iterative technique for the rectification of observed distributions,” Astron. J. 79, 745–754 (1974).
[CrossRef]

IEEE Trans. Med. Imag. (2)

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

L. Kaufman, “Implementing and accelerating the EM algorithm for positron emission tomography,”IEEE Trans. Med. Imag. MI-6, 37–51 (1987).
[CrossRef]

IEEE Trans. Nucl. Sci. (1)

D. L. Snyder, D. G. Politte, “Image reconstruction from list-mode data in an emission tomography system having time-of-flight measurements,”IEEE Trans. Nucl. Sci. NS-30, 1843–1849 (1983).
[CrossRef]

J. Am. Stat. Assoc. (1)

Y. Vardi, L. A. Shepp, L. Kaufman, “A statistical model for positron emission tomography,”J. Am. Stat. Assoc. 80, 8–38 (1985).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

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

A. P. Dempster, N. M. Laird, D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,”J. R. Stat. Soc. Ser. B 39, 1–38 (1977).

Opt. Eng. (2)

J. D. Gonglewski, C. R. DeHainaut, C. M. Lampkin, R. C. Dymale, “System design of a wavefront sensing package for a wide field of view optical phased array,” Opt. Eng. 27, 785–792 (1988).
[CrossRef]

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

Sov. Phys. JETP (1)

N. B. Baranova, B. Y. Zel’dovich, “Dislocations of the wave-front surface and zeros of the amplitude,” Sov. Phys. JETP 53, 925–929 (1981).

Other (10)

D. C. Luenberger, Linear and Nonlinear Programming (Addison-Wesley, Reading, Mass., 1984).

D. G. Politte, “Reconstruction algorithms for time-of-flight assisted positron-emission tomographs,” M.S. thesis (Sever Institute of Technology, Washington University, St. Louis, Mo., 1983).

R. G. Paxman, S. L. Crippen, “Aberration correction for phased-array telescopes using phase diversity,” in Digital Image Synthesis and Inverse Optics, A. F. Gmitro, P. S. Idell, I. J. LaHaie, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1351, 787–797 (1990).
[CrossRef]

J. M. Mendel, Lessons in Digital Estimation Theory (Prentice-Hall, Englewood Cliffs, N.J., 1987).

A. V. Oppenheim, R. W. Schafer, Discrete-Time Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1989).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968).

F. Roddier, “Passive versus active methods in optical interferometry,” in Proceedings of the ESO/NOAO Conference on High Resolution Imaging by Interferometry, F. Merkle, ed. (European Southern Observatory, Garching, Federal Republic of Germany, 1988), pp. 565–574.

R. G. Lyon, “HST phase retrieval: a parameter estimation,” in Applications of Digital Image Processing XIV, A. G. Tescher, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1567, 317–326 (1991).
[CrossRef]

J. R. Fienup, “Phase retrieval for the Hubble Space Telescope using iterative propagation algorithms,” in Applications of Digital Image Processing XIV, A. G. Tescher, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1567, 327–332 (1991).
[CrossRef]

R. A. Gonsalves, R. Childlaw, “Wavefront sensing by phase retrieval,” in Applications of Digital Image Processing III, A. G. Tescher, ed., Proc. Soc. Photo-Opt. Instrum. Eng.207, 32–39 (1979).
[CrossRef]

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Figures (2)

Fig. 1
Fig. 1

Optical layout of a phase-diversity system. The conventional image is degraded by aberrations in the optical system. The diversity image is degraded by the combination of the same aberrations and a known amount of defocus.

Fig. 2
Fig. 2

Pictorial representation of the construction of the aberration-only objective function. The two-dimensional contour plot represents the log-likelihood function. A ridge of the contour plot is defined by the locus of points (X, YM) for which YM maximizes L for each value of X. The projection of the ridge values in the Y direction yields the one-dimensional, aberration-only objective function.

Equations (146)

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g k ( x ) = x χ f ( x ) s k ( x - x )
f * s k ( x ) ,
χ = { 0 , 1 , , N - 1 } × { 0 , 1 , , N - 1 } .
H k ( u ) = H k ( u ) exp { i [ ϕ ( u ) + θ k ( u ) ] } ,
ϕ ( u ) = j = 1 J α j ϕ j ( u ) ,
ϕ j ( u ) = δ u u j { 1 u = u j 0 u u j .
h k ( x ) = 1 N 2 u χ H k ( u ) exp ( i 2 π u , x / N ) ,
s k ( x ) = h k ( x ) 2 .
d k ( x ) = g k ( x ) + n k ( x )
= f * s k ( x ) + n k ( x ) ,
p [ d k ( x ) ; f , α ] = 1 ( 2 π σ n 2 ) 1 / 2 exp { - [ d k ( x ) - f * s k ( x ) ] 2 2 σ n 2 } .
p ( { d k } ; f , α ) = k = 1 K x χ 1 ( 2 π σ n 2 ) 1 / 2 × exp { - [ d k ( x ) - f * s k ( x ) ] 2 2 σ n 2 } .
L ( f , α ) = - k = 1 K x χ [ d k ( x ) - f * s k ( x ) ] 2 ,
L ( f , α ) = - 1 N 2 k = 1 K u χ D k ( u ) - F ( u ) S k ( u ) 2 ,
F M ( u ) = D 1 ( u ) S 1 * ( u ) + D 2 ( u ) S 2 * ( u ) S 1 ( u ) 2 + S 2 ( u ) 2 ,
L M ( α ) = - u χ D 1 ( u ) S 2 ( u ) - D 2 ( u ) S 1 ( u ) 2 S 1 ( u ) 2 + S 2 ( u ) 2 ,
F r ( u ) L = 0 ,
F i ( u ) L = 0 ,             u χ ,
F M ( u ) = { k = 1 K D k ( u ) S k * ( u ) / l = 1 K S l ( u ) 2 u χ 1 F M * ( - u ) u χ 0 ,
L M ( α ) = - u χ 1 j = 1 K - 1 k = j + 1 K D j ( u ) S k ( u ) - D k ( u ) S j ( u ) 2 l = 1 K S l ( u ) 2 - u χ 0 k = 1 K D k ( u ) 2 .
L M ( α ) = u χ 1 | j = 1 K D j ( u ) S j * ( u ) | 2 l = 1 K S l ( u ) 2 - u χ k = 1 K D k ( u ) 2 .
α n L M = - 4 N 2 u χ ϕ n ( u ) Im [ k = 1 K H k ( u ) ( Z k * H k * ) ( u ) ] ,
Z k ( u ) { [ l S l 2 ( j D j S j * ) D k * - | j D j S j * | 2 S k * ] / ( l S l 2 ) 2 u χ 1 0 u χ 0 .
α n L M = 4 N 2 u χ ϕ n ( u ) Im [ H 1 ( u ) ( Z S 2 * H 1 * ) ( u ) - H 2 ( u ) ( Z S 1 * H 2 * ) ( u ) ] ,
Z ( u ) { ( D 1 S 1 * + D 2 S 2 * ) ( D 2 * S 1 * - D 1 * S 2 * ) ( S 1 2 + S 2 2 ) 2 u χ 1 0 u χ 0 .
α n L m = - 4 N 2 Im [ k = 1 K H k ( u n ) ( Z k * H k * ) ( u n ) ] .
Pr [ d k ( x ) ] = g k ( x ) d k ( x ) exp [ - g k ( x ) ] d k ( x ) ! .
Pr ( { d k } ) = k = 1 K x χ g k ( x ) d k ( x ) exp [ - g k ( x ) ] d k ( x ) ! .
L ( f , α ) = k = 1 K x χ [ d k ( x ) ln g k ( x ) - g k ( x ) ] ,
k = 1 K x χ g k ( x ) = k = 1 K x χ x χ f ( x ) s k ( x - x )
= k = 1 K x χ f ( x ) x χ s k ( x - x )
= k = 1 K x χ f ( x ) x χ h k ( x - x ) 2
= k = 1 K x χ f ( x ) 1 N 2 × u χ H k ( u ) exp ( - i 2 π u , x / N ) 2
= x χ f ( x ) 1 N 2 k = 1 K u χ H k ( u ) 2 ,
C 1 N 2 k = 1 K u χ H k ( u ) 2 .
k = 1 K x χ g k ( x ) = C x χ f ( x ) ,
L ( f , α ) = k = 1 K x χ d k ( x ) ln g k ( x ) - C x χ f ( x )
= k = 1 K x χ d k ( x ) ln [ x χ f ( x ) s k ( x - x ) ] - C x χ f ( x ) .
f ( x i ) L = k = 1 K x χ d k ( x ) f ( x i ) ln [ x χ f ( x ) s k ( x - x ) ] - C x χ f ( x i ) f ( x )
= k = 1 K x χ d k ( x ) s k ( x - x i ) x χ f ( x ) s k ( x - x ) - C .
0 = k = 1 K x χ d k ( x ) s k ( x - x i ) x χ f ( x ) s k ( x - x ) - C .
α n L = - 2 u χ ϕ n ( u ) Im [ k = 1 K H k ( u ) 1 N 2 x χ h k * ( x ) × exp ( i 2 π u , x / N x χ d k ( x ) f ( x - x ) x χ f ( x ) s k ( x - x ) ] .
L w ( f , ϕ ) = k = 1 K x χ [ d k ( x ) ln g k ( x ) - g k ( x ) ] w k ( x ) ,
f ( x i ) L w = k = 1 K x χ [ d k ( x ) g k ( x ) - 1 ] w k ( x ) s k ( x - x i ) .
a n L w = - 2 u χ ϕ n ( u ) Im { k = 1 K H k ( u ) 1 N 2 x χ h k * ( x ) × exp ( i 2 π u , x ) / N ) x χ [ d k ( x ) g k ( x ) - 1 ] w k ( x ) f ( x - x ) } .
f ( x ) = a δ ( x ) ,
δ ( x ) { 1 x = ( 0 , 0 ) 0 otherwise .
α n L = - 2 u χ ϕ n ( u ) Im [ k = 1 K H k ( u ) 1 N 2 x χ h k * ( x ) d k ( x ) s k ( x ) × exp ( i 2 π u , x / N ) ] .
F u - 1 [ h k * ( x ) d k ( x ) s k ( x ) ] 1 N 2 x χ h k * ( x ) d k ( x ) s k ( x ) exp ( i 2 π u , x / N ) ,
α n L = - 2 u χ ϕ n ( u ) Im { k = 1 K H k ( u ) F u - 1 [ h k * ( x ) d k ( x ) s k ( x ) ] } .
g k ( x ) = x χ f ( x ) s k ( x - x ) .
g ˜ k ( x x ) = f ( x ) s k ( x - x ) .
x χ d ˜ k ( x x ) ,
x χ g ˜ k ( x x ) = x χ f ( x ) s k ( x - x )
= g k ( x ) .
d k ( x ) = x χ d ˜ k ( x x ) .
L [ f ( r + 1 ) , α ( r + 1 ) ] L [ f ( r ) , α ( r ) ] ,
L c d ( f , α ) = k = 1 K x χ x χ { d ˜ k ( x x ) × ln [ f ( x ) s k ( x - x ) ] - f ( x ) s k ( x - x ) }
= x χ [ k = 1 K x χ d ˜ k ( x x ) ] ln f ( x ) + k = 1 K x χ x χ d ˜ k ( x x ) ln s k ( x - x ) - x χ f ( x ) k = 1 K x χ s k ( x - x )
= x χ [ k = 1 K x χ d ˜ k ( x x ) ] ln f ( x ) + k = 1 K x χ [ x χ d ˜ k ( x x - x ) ] ln s k ( x ) - x χ f ( x ) k = 1 K x χ s k ( x ) ,
Q [ f , α f ( r ) , α ( r ) ] E ( r ) [ L c d ( f , α ) { d k } ] ,
f ( r + 1 ) ( x ) = f ( r ) ( x ) 1 k = 1 K S k ( 0 ) k = 1 K x χ s k ( r ) ( x - x ) d k ( x ) g k ( r ) ( x ) ,
α ( r + 1 ) = arg max α { k = 1 K x χ × [ x χ f ( r ) ( x - x ) s k ( r ) ( x ) g k ( r ) ( x ) d k ( x ) ] ln s k ( x ) } ,
g k ( r ) ( x ) x χ f k ( r ) ( x ) s k ( r ) ( x - x ) ,
s k ( r ) k th diversity PSF evaluated for α ( r ) ,
S k ( 0 ) = x χ s k ( x ) .
f ( r + 1 ) ( x ) = f ( r ) ( x ) 1 k = 1 K S k ( 0 ) k = 1 K x χ s k ( x - x ) d k ( x ) g k ( r ) ( x ) ,
L ( f , α ) = - 1 N 2 k = 1 K u χ D k ( u ) - F ( u ) S k ( u ) 2 .
F r ( u ) L = 0 ,
F i ( u ) L = 0 ,             u χ ,
F r L = - 1 N 2 k = 1 K F r ( D k 2 + F S k 2 - D k F * S k * - D k * F S k )
= - 1 N 2 k = 1 K [ S k 2 F r ( F r 2 + F i 2 ) - D k S k * - D k * S k ]
= - 1 N 2 k = 1 K ( 2 S k 2 F r - D k S k * - D k * S k ) ,
F i L = - 1 N 2 k = 1 K ( 2 S k 2 F i + i D k S k * - i D k * S k ) .
2 F r k = 1 K S k 2 = k = 1 K ( D k S k * + D k * S k ) ,
F r ( u ) = { k = 1 K ( D k S k * + D k * S k ) / 2 l = 1 K S l 2 u χ 1 F r ( - u ) u χ 0 ,
χ 0 = { u : S k ( u ) = 0 , k = 1 , , K } ,
χ 1 = { u : u χ , u χ 0 } .
F i ( u ) = { k = 1 K ( - D k S k * + D k * S k ) i / 2 l = 1 K S l 2 u χ 1 - F i ( - u ) u χ 0 .
F M ( u ) = F r ( u ) + i F i ( u )
= { k = 1 K [ D k S k * + D k * S k + i ( - D k S k * i + D k * S k i ) / 2 l = 1 K S l 2 u χ 1 F M * ( - u ) u χ 0 ,
= { k = 1 K D k S k * / l = 1 K S l 2 u χ 1 F M * ( - u ) u χ 0
L M ( α ) = - u χ k = 1 K D k ( u ) - F M ( u ) S k ( u ) 2
= - u χ 1 k | D k - S k j D j S j * l S j 2 | 2 - u χ 0 k D k 2 ,
L M 1 = - u χ 1 k | D k l S l 2 - S k j D j S j * l S l 2 | 2
= - u χ 1 k | D k l S l 2 | 2 + | S k j D j S j * | 2 - D k * l S l 2 S k j D j S j * - D k l S l 2 S k * j D j * S j ( l S l 2 ) 2
numerator = k D k 2 ( l S l 2 ) 2 + k S k 2 | j D j S j * | 2 - k D k * S k l S l 2 j D j S j * - k D k S k * l S l 2 j D j * S j
= k D k 2 ( l S l 2 ) 2 - l S l 2 | j D j S j * | 2 ,
L M 1 = - u χ 1 k D k 2 j S j 2 - | j D j S j * | 2 l S l 2
= - u χ 1 k j D k S j 2 - j k D j S j * D k * S k l S l 2
= - u χ 1 ( 1 / 2 ) k j D k S j - D j S k 2 l S l 2 ,
L M 1 = - u χ 1 j = 1 K - 1 k = j + 1 K D j ( u ) S k ( u ) - D k ( u ) S j ( u ) 2 l = 1 K S l ( u ) 2 .
L M ( α ) = - u χ 1 j = 1 K - 1 k = j + 1 K D j ( u ) S k ( u ) - D k ( u ) S j ( u ) 2 l = 1 K S l ( u ) 2 - u χ 0 k = 1 K D k ( u ) 2 .
L M ( α ) = - u χ 1 [ k = 1 K D k ( u ) 2 - | j = 1 K D j ( u ) S j * ( u ) | 2 l = 1 K S l ( u ) 2 ] - u χ 0 k = 1 K D k ( u ) 2
= u χ 1 | j = 1 K D j ( u ) S j * ( u ) | 2 l = 1 K S l ( u ) 2 - u χ k = 1 K D k ( u ) 2 .
α n L M = α n u χ 1 | j = 1 K D j ( u ) S j * ( u ) | 2 l = 1 K S l ( u ) 2 - α n u χ k = 1 K D k ( u ) 2 .
α n L M = u χ 1 α n | j = 1 K D j ( u ) S j * ( u ) | 2 l = 1 K S l ( u ) 2
= u χ 1 l S l 2 ( j D j S j * k D k * S k + c . c . ) - | j D j S j * | 2 ( l S l * S l + c . c . ) ( l S l 2 ) 2
= u χ 1 l S l 2 j D j S j * k D k * S k - | j D j S j * | 2 l S l * S l ( l S l 2 ) 2 + c . c .
= u χ k Z k S k + c . c . ,
Z k ( u ) { [ l S l 2 ( j D j S j * ) D k * - | j D j S j * | 2 S k * ] / ( l S l 2 ) 2 u χ l 0 u χ 0 .
S k ( u ) = x χ s k ( x ) exp ( - i 2 π u , x / N )
= x χ h k ( x ) 2 exp ( - i 2 π u , x / N ) .
S k ( u ) = 1 N 2 u χ H k ( u ) H k * ( u - u ) ,
H k ( u ) H k ( u ) exp { i [ θ k + j = 1 J α j ϕ j ( u ) ] } .
α n S k ( u ) = 1 N 2 u χ [ H k * ( u - u ) α n H k ( u ) + H k ( u ) α n H k * ( u - u ) ]
= ( 1 / N 2 ) u χ [ H k * ( u - u ) i ϕ n ( u ) H k ( u ) - H k ( u ) i ϕ n ( u - u ) H k * ( u - u ) ]
= 1 N 2 [ u χ i ϕ n ( u ) H k ( u ) H k * ( u - u ) - u χ i ϕ n ( u ) H k ( u + u ) H k * ( u ) ]
= ( i / N 2 ) u χ ϕ n ( u ) [ H k ( u ) H k * ( u - u ) - H k * ( u ) H k ( u + u ) ] ,
α n L M = u χ k = 1 K { Z k ( u ) i N 2 u χ ϕ n ( u ) × [ H k ( u ) H k * ( u - u ) - H k * ( u ) H k ( u + u ) ] } + c . c .
= [ i N 2 u χ ϕ n ( u ) k = 1 K H k ( u ) u χ Z k ( u ) H k * ( u - u ) ] + c . c . - [ i N 2 u χ ϕ n ( u ) k = 1 K H k * ( u ) u χ Z k ( u ) H k ( u + u ) ] + c . c .
second term = - [ i N 2 u χ ϕ n ( u ) k = 1 K H k * ( u ) × u χ Z k * ( - u ) H k ( u + u ) ] + c . c .
= - [ i N 2 u χ ϕ n ( u ) k = 1 K H k * ( u ) × u χ Z k * ( u ) H k ( u - u ) ] + c . c . ,
α n L M = [ 2 i N 2 u χ ϕ n ( u ) k = 1 K H k ( u ) × u χ Z k ( u ) H k * ( u - u ) ] + c . c .
= - 4 N 2 u χ ϕ n ( u ) Im [ k = 1 K H k ( u ) ( Z k * H k * ) ( u ) ] ,
α n L = k = 1 K x χ d k ( x ) x f ( x ) s k ( x - x ) x χ f ( x ) α n s k ( x - x )
= k = 1 K x χ d k ( x ) g k ( x ) x χ f ( x ) α n s k ( x - x ) ,
α n s k ( x ) = h k * ( x ) h k ( x ) α n + c . c .
= h k * ( x ) N 2 α n u χ H k ( u ) exp ( i 2 π u , x / N ) + c . c .
= h k * ( x ) N 2 α n u χ H k ( u ) × exp { i [ θ k ( u ) + j = 1 J α j ϕ j ( u ) ] } × exp ( i 2 π u , x / N ) + c . c .
= h k * ( x ) N 2 u χ i ϕ n ( u ) H k ( u ) exp ( i 2 π u , x / N ) + c . c . ,
α n L = k = 1 K x χ d k ( x ) g k ( x ) x χ f ( x ) [ h k * ( x - x ) N 2 u χ i ϕ n ( u ) H k ( u ) exp ( i 2 π u , x - x / N ) + c . c . ]
= [ k = 1 K x χ d k ( x ) g k ( x ) x χ f ( x ) h k * ( x - x ) N 2 u χ i ϕ n ( u ) H k ( u ) exp ( i 2 π u , x - x / N ) ] + c . c .
= [ k = 1 K u χ i ϕ n ( u ) H k ( u ) 1 N 2 x χ d k ( x ) g k ( x ) x χ f ( x ) h k * ( x - x ) exp ( i 2 π u , x - x / N ) ] + c . c .
= [ k = 1 K u χ i ϕ n ( u ) H k ( u ) 1 N 2 x χ d k ( x ) g k ( x ) x χ f ( x - x ) h k * ( x ) exp ( i 2 π u , x / N ) ] + c . c .
= [ k = 1 K u χ i ϕ n ( u ) H k ( u ) 1 N 2 x χ h k * ( x ) exp ( i 2 π u , x / N ) x χ d k ( x ) f ( x - x ) g k ( x ) ] + c . c .
= - 2 Im [ k = 1 K u χ ϕ n ( u ) H k ( u ) 1 N 2 x χ h k * ( x ) exp ( i 2 π u , x / N ) x χ d k ( x ) f ( x - x ) g k ( x ) ]
= - 2 u χ ϕ n ( u ) Im [ k = 1 K H k ( u ) 1 N 2 x χ h k * ( x ) exp ( i 2 π u , x / N ) x χ d k ( x ) f ( x - x ) x f ( x ) s k ( x - x ) ] ,
Q [ f , α f ( r ) , α ( r ) ] = E ( r ) [ L c d ( f , α ) { d k } ] ,
L c d ( f , α ) = x χ [ k = 1 K x χ d ˜ k ( x x ) ] ln f ( x ) + k = 1 K x χ [ x χ d ˜ k ( x x - x ) ] ln s k ( x ) - x χ f ( x ) k = 1 K x χ s k ( x ) .
E ( r ) [ d ˜ k ( x x ) { d k } ] = E ( r ) [ d ˜ k ( x x ) d k ( x ) ] ,
d k ( x ) = x χ d ˜ k ( x x ) .
E [ z 1 z 1 + z 2 ] = λ 1 λ 1 + λ 2 ( z 1 + z 2 ) .
E ( r ) [ d ˜ k ( x x ) d k ( x ) ] = f ( r ) ( x ) s k ( r ) ( x - x ) g k ( r ) ( x ) d k ( x ) ,
g k ( r ) ( x ) x χ f k ( r ) ( x ) s k ( r ) ( x - x ) ,
s k ( r ) k th diversity PSF evaluated for α ( r ) .
Q [ f , α f ( r ) , α ( r ) ] = x χ [ k = 1 K x χ f ( r ) ( x ) s k ( r ) ( x - x ) g k ( r ) ( x ) d k ( x ) ] ln f ( x ) + k = 1 K x χ [ x χ f ( r ) ( x - x ) s k ( r ) ( x ) g k ( r ) ( x ) d k ( x ) ] ln s k ( x ) - x χ f ( x ) k = 1 K S k ( 0 ) ,
Q [ f , α f ( r ) , α ( r ) ] = Q f + Q α ,
Q f x χ [ k = 1 K x χ f ( r ) ( x ) s k ( r ) ( x - x ) g k ( r ) ( x ) d k ( x ) ] ln f ( x ) - x χ f ( x ) k = 1 K S k ( 0 ) ,
Q α k = 1 K x χ [ x χ f ( r ) ( x - x ) s k ( r ) ( x ) g k ( r ) ( x ) d k ( x ) ] ln s k ( x ) .
Q f f ( x ) = 1 f ( x ) k = 1 K x χ f ( r ) ( x ) s k ( r ) ( x - x ) g k ( r ) ( x ) d k ( x ) - k = 1 K S k ( 0 ) ,
2 Q f f ( x ) f ( x ) = - δ ( x - x ) 1 [ f ( x ) ] 2 × k = 1 K x χ f ( r ) ( x ) s k ( r ) ( x - x ) g k ( r ) ( x ) d k ( x ) 0 ,
f ( r + 1 ) ( x ) = f ( r ) ( x ) 1 k = 1 K S k ( 0 ) k = 1 K x χ s k ( r ) ( x - x ) d k ( x ) g k ( r ) ( x ) .
α ( r + 1 ) = arg max α { k = 1 K x χ [ x χ f ( r ) ( x - x ) s k ( r ) ( x ) g k ( r ) ( x ) d k ( x ) ] × ln s k ( x ) } .
L ( α ) = k = 1 K x χ d k ( x ) ln s k ( x ) + C .
d k ( x ) x χ f ( r ) ( x - x ) s k ( r ) ( x ) g k ( r ) ( x ) d k ( x )

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