Abstract

We propose a novel method called marginal estimator for estimating the aberrations and the object from phase-diversity data. The conventional estimator found in the literature concerning the technique first proposed by Gonsalves has its basis in a joint estimation of the aberrated phase and the observed object. By means of simulations, we study the behavior of the conventional estimator, which is interpretable as a joint maximum a posteriori approach, and we show in particular that it has undesirable asymptotic properties and does not permit an optimal joint estimation of the object and the aberrated phase. We propose a novel marginal estimator of the sole phase by maximum a posteriori. It is obtained by integrating the observed object out of the problem. This reduces drastically the number of unknowns, allows the unsupervised estimation of the regularization parameters, and provides better asymptotic properties. We show that the marginal method is also appropriate for the restoration of the object. This estimator is implemented and its properties are validated by simulations. The performance of the joint method and the marginal one is compared on both simulated and experimental data in the case of Earth observation. For the studied object, the comparison of the quality of the phase restoration shows that the performance of the marginal approach is better under high-noise-level conditions.

© 2003 Optical Society of America

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References

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  1. R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21, 829–832 (1982).
    [CrossRef]
  2. R. L. Kendrick, D. S. Acton, A. L. Duncan, “Phase-diversity wave-front sensor for imaging systems,” Appl. Opt. 33, 6533–6546 (1994).
    [CrossRef] [PubMed]
  3. D. J. Lee, M. C. Roggemann, B. M. Welsh, E. R. Crosby, “Evaluation of least-squares phase-diversity technique for space telescope wave-front sensing,” Appl. Opt. 36, 9186–9197 (1997).
    [CrossRef]
  4. M. G. Löfdahl, A. L. Duncan, “Fast phase diversity wavefront sensor for mirror control,” in Adaptative Optical System Technologies, D. Bonaccini, R. K. Tyson, eds., Proc. SPIE3353, 952–963 (1998).
    [CrossRef]
  5. M. G. Löfdahl, G. B. Scharmer, “Wavefront sensing and image restoration from focused and defocused solar images,” Astron. Astrophys. 107, 243–264 (1994).
  6. J. H. Seldin, R. G. Paxman, “Phase-diverse speckle reconstruction of solar data,” in Image Reconstruction and Restoration, T. J. Schulz, D. L. Snyder, eds., Proc. SPIE2302, 268–280 (1994).
    [CrossRef]
  7. B. J. Thelen, R. G. Paxman, D. A. Carrara, J. H. Seldin, “Maximum a posteriori estimation of fixed aberrations, dynamic aberrations, and the object from phase-diverse speckle data,” J. Opt. Soc. Am. A 16, 1016–1025 (1999).
    [CrossRef]
  8. O. M. Bucci, A. Capozzoli, G. D’Elia, “Regularizing strategy for image restoration and wave-front sensing by phase diversity,” J. Opt. Soc. Am. A 16, 1759–1768 (1999).
    [CrossRef]
  9. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976).
    [CrossRef]
  10. 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]
  11. A. P. Kattnig, J. Primot, “Model of the second-order statistic of the radiance field of natural scenes, adapted to system conceiving,” in Visual Information Processing VI, S. K. Park, R. D. Juday, eds., Proc. SPIE3074, 132–141 (1997).
    [CrossRef]
  12. R. J. A. Little, D. B. Rubin, “On Jointly Estimating Parameters and Missing Data by Maximizing the Complete-Data Likelihood,” Am. Stat. 37, 218–220 (1983).
  13. Y. Goussard, G. Demoment, J. Idier, “A new algorithm for iterative deconvolution of sparse spike,” in Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1990), pp. 1547–1550.
  14. F. Champagnat, J. Idier, “An alternative to standard maximum likelihood for Gaussian mixtures,” in Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1995), pp. 2020–2023.
  15. E. Lehmann, Theory of Point Estimation (Wiley, New York1983).
  16. E. D. Carvalho, D. Slock, “Maximum-likelihood blind FIR multi-channel estimation with Gaussian prior for the symbols,” in Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1997), pp. 3593–3596.
  17. L. Meynadier, V. Michau, M.-T. Velluet, J.-M. Conan, L. M. Mugnier, G. Rousset, “Noise propagation in wave-front sensing with phase diversity,” Appl. Opt. 38, 4967–4979 (1999).
    [CrossRef]
  18. J. Seldin, R. Paxman, “Closed-loop wavefront sensing for a sparse-aperture, phased-array telescope using broadband phase diversity,” in Imaging Technology and Telescopes, J. W. Bilbro, J. B. Breckinridge, R. A. Carreras, S. R. Czyzak, M. J. Eckart, R. D. Fiete, P. S. Idell, eds., Proc. SPIE4091, 48–63 (2000).
    [CrossRef]
  19. D. A. Carrara, B. J. Thelen, R. G. Paxman, “Aberration correction of segmented-aperture telescopes by using phase diversity,” in Image Reconstruction from Incomplete Data, M. A. Fiddy, R. P. Millane, eds., Proc. SPIE4123, 56–63 (2000).
    [CrossRef]
  20. O. N. Strand, “Theory and methods related to the singular-function expansion and Landweber’s iteration for integral equations of the first kind,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 11, 798–825 (1974).
    [CrossRef]
  21. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, U.K., 1983).
  22. F. R. Gantmacher, Théorie des Matrices Tome I (Dunod, Paris1966).

1999 (3)

1997 (1)

1994 (2)

R. L. Kendrick, D. S. Acton, A. L. Duncan, “Phase-diversity wave-front sensor for imaging systems,” Appl. Opt. 33, 6533–6546 (1994).
[CrossRef] [PubMed]

M. G. Löfdahl, G. B. Scharmer, “Wavefront sensing and image restoration from focused and defocused solar images,” Astron. Astrophys. 107, 243–264 (1994).

1992 (1)

1983 (1)

R. J. A. Little, D. B. Rubin, “On Jointly Estimating Parameters and Missing Data by Maximizing the Complete-Data Likelihood,” Am. Stat. 37, 218–220 (1983).

1982 (1)

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

1976 (1)

1974 (1)

O. N. Strand, “Theory and methods related to the singular-function expansion and Landweber’s iteration for integral equations of the first kind,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 11, 798–825 (1974).
[CrossRef]

Acton, D. S.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, U.K., 1983).

Bucci, O. M.

Capozzoli, A.

Carrara, D. A.

B. J. Thelen, R. G. Paxman, D. A. Carrara, J. H. Seldin, “Maximum a posteriori estimation of fixed aberrations, dynamic aberrations, and the object from phase-diverse speckle data,” J. Opt. Soc. Am. A 16, 1016–1025 (1999).
[CrossRef]

D. A. Carrara, B. J. Thelen, R. G. Paxman, “Aberration correction of segmented-aperture telescopes by using phase diversity,” in Image Reconstruction from Incomplete Data, M. A. Fiddy, R. P. Millane, eds., Proc. SPIE4123, 56–63 (2000).
[CrossRef]

Carvalho, E. D.

E. D. Carvalho, D. Slock, “Maximum-likelihood blind FIR multi-channel estimation with Gaussian prior for the symbols,” in Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1997), pp. 3593–3596.

Champagnat, F.

F. Champagnat, J. Idier, “An alternative to standard maximum likelihood for Gaussian mixtures,” in Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1995), pp. 2020–2023.

Conan, J.-M.

Crosby, E. R.

D’Elia, G.

Demoment, G.

Y. Goussard, G. Demoment, J. Idier, “A new algorithm for iterative deconvolution of sparse spike,” in Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1990), pp. 1547–1550.

Duncan, A. L.

R. L. Kendrick, D. S. Acton, A. L. Duncan, “Phase-diversity wave-front sensor for imaging systems,” Appl. Opt. 33, 6533–6546 (1994).
[CrossRef] [PubMed]

M. G. Löfdahl, A. L. Duncan, “Fast phase diversity wavefront sensor for mirror control,” in Adaptative Optical System Technologies, D. Bonaccini, R. K. Tyson, eds., Proc. SPIE3353, 952–963 (1998).
[CrossRef]

Fienup, J. R.

Gantmacher, F. R.

F. R. Gantmacher, Théorie des Matrices Tome I (Dunod, Paris1966).

Gonsalves, R. A.

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

Goussard, Y.

Y. Goussard, G. Demoment, J. Idier, “A new algorithm for iterative deconvolution of sparse spike,” in Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1990), pp. 1547–1550.

Idier, J.

Y. Goussard, G. Demoment, J. Idier, “A new algorithm for iterative deconvolution of sparse spike,” in Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1990), pp. 1547–1550.

F. Champagnat, J. Idier, “An alternative to standard maximum likelihood for Gaussian mixtures,” in Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1995), pp. 2020–2023.

Kattnig, A. P.

A. P. Kattnig, J. Primot, “Model of the second-order statistic of the radiance field of natural scenes, adapted to system conceiving,” in Visual Information Processing VI, S. K. Park, R. D. Juday, eds., Proc. SPIE3074, 132–141 (1997).
[CrossRef]

Kendrick, R. L.

Lee, D. J.

Lehmann, E.

E. Lehmann, Theory of Point Estimation (Wiley, New York1983).

Little, R. J. A.

R. J. A. Little, D. B. Rubin, “On Jointly Estimating Parameters and Missing Data by Maximizing the Complete-Data Likelihood,” Am. Stat. 37, 218–220 (1983).

Löfdahl, M. G.

M. G. Löfdahl, G. B. Scharmer, “Wavefront sensing and image restoration from focused and defocused solar images,” Astron. Astrophys. 107, 243–264 (1994).

M. G. Löfdahl, A. L. Duncan, “Fast phase diversity wavefront sensor for mirror control,” in Adaptative Optical System Technologies, D. Bonaccini, R. K. Tyson, eds., Proc. SPIE3353, 952–963 (1998).
[CrossRef]

Meynadier, L.

Michau, V.

Mugnier, L. M.

Noll, R. J.

Paxman, R.

J. Seldin, R. Paxman, “Closed-loop wavefront sensing for a sparse-aperture, phased-array telescope using broadband phase diversity,” in Imaging Technology and Telescopes, J. W. Bilbro, J. B. Breckinridge, R. A. Carreras, S. R. Czyzak, M. J. Eckart, R. D. Fiete, P. S. Idell, eds., Proc. SPIE4091, 48–63 (2000).
[CrossRef]

Paxman, R. G.

B. J. Thelen, R. G. Paxman, D. A. Carrara, J. H. Seldin, “Maximum a posteriori estimation of fixed aberrations, dynamic aberrations, and the object from phase-diverse speckle data,” J. Opt. Soc. Am. A 16, 1016–1025 (1999).
[CrossRef]

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. H. Seldin, R. G. Paxman, “Phase-diverse speckle reconstruction of solar data,” in Image Reconstruction and Restoration, T. J. Schulz, D. L. Snyder, eds., Proc. SPIE2302, 268–280 (1994).
[CrossRef]

D. A. Carrara, B. J. Thelen, R. G. Paxman, “Aberration correction of segmented-aperture telescopes by using phase diversity,” in Image Reconstruction from Incomplete Data, M. A. Fiddy, R. P. Millane, eds., Proc. SPIE4123, 56–63 (2000).
[CrossRef]

Primot, J.

A. P. Kattnig, J. Primot, “Model of the second-order statistic of the radiance field of natural scenes, adapted to system conceiving,” in Visual Information Processing VI, S. K. Park, R. D. Juday, eds., Proc. SPIE3074, 132–141 (1997).
[CrossRef]

Roggemann, M. C.

Rousset, G.

Rubin, D. B.

R. J. A. Little, D. B. Rubin, “On Jointly Estimating Parameters and Missing Data by Maximizing the Complete-Data Likelihood,” Am. Stat. 37, 218–220 (1983).

Scharmer, G. B.

M. G. Löfdahl, G. B. Scharmer, “Wavefront sensing and image restoration from focused and defocused solar images,” Astron. Astrophys. 107, 243–264 (1994).

Schulz, T. J.

Seldin, J.

J. Seldin, R. Paxman, “Closed-loop wavefront sensing for a sparse-aperture, phased-array telescope using broadband phase diversity,” in Imaging Technology and Telescopes, J. W. Bilbro, J. B. Breckinridge, R. A. Carreras, S. R. Czyzak, M. J. Eckart, R. D. Fiete, P. S. Idell, eds., Proc. SPIE4091, 48–63 (2000).
[CrossRef]

Seldin, J. H.

B. J. Thelen, R. G. Paxman, D. A. Carrara, J. H. Seldin, “Maximum a posteriori estimation of fixed aberrations, dynamic aberrations, and the object from phase-diverse speckle data,” J. Opt. Soc. Am. A 16, 1016–1025 (1999).
[CrossRef]

J. H. Seldin, R. G. Paxman, “Phase-diverse speckle reconstruction of solar data,” in Image Reconstruction and Restoration, T. J. Schulz, D. L. Snyder, eds., Proc. SPIE2302, 268–280 (1994).
[CrossRef]

Slock, D.

E. D. Carvalho, D. Slock, “Maximum-likelihood blind FIR multi-channel estimation with Gaussian prior for the symbols,” in Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1997), pp. 3593–3596.

Strand, O. N.

O. N. Strand, “Theory and methods related to the singular-function expansion and Landweber’s iteration for integral equations of the first kind,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 11, 798–825 (1974).
[CrossRef]

Thelen, B. J.

B. J. Thelen, R. G. Paxman, D. A. Carrara, J. H. Seldin, “Maximum a posteriori estimation of fixed aberrations, dynamic aberrations, and the object from phase-diverse speckle data,” J. Opt. Soc. Am. A 16, 1016–1025 (1999).
[CrossRef]

D. A. Carrara, B. J. Thelen, R. G. Paxman, “Aberration correction of segmented-aperture telescopes by using phase diversity,” in Image Reconstruction from Incomplete Data, M. A. Fiddy, R. P. Millane, eds., Proc. SPIE4123, 56–63 (2000).
[CrossRef]

Velluet, M.-T.

Welsh, B. M.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, U.K., 1983).

Am. Stat. (1)

R. J. A. Little, D. B. Rubin, “On Jointly Estimating Parameters and Missing Data by Maximizing the Complete-Data Likelihood,” Am. Stat. 37, 218–220 (1983).

Appl. Opt. (3)

Astron. Astrophys. (1)

M. G. Löfdahl, G. B. Scharmer, “Wavefront sensing and image restoration from focused and defocused solar images,” Astron. Astrophys. 107, 243–264 (1994).

J. Opt. Soc. Am. (1)

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

Opt. Eng. (1)

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

SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. (1)

O. N. Strand, “Theory and methods related to the singular-function expansion and Landweber’s iteration for integral equations of the first kind,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 11, 798–825 (1974).
[CrossRef]

Other (11)

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, U.K., 1983).

F. R. Gantmacher, Théorie des Matrices Tome I (Dunod, Paris1966).

J. H. Seldin, R. G. Paxman, “Phase-diverse speckle reconstruction of solar data,” in Image Reconstruction and Restoration, T. J. Schulz, D. L. Snyder, eds., Proc. SPIE2302, 268–280 (1994).
[CrossRef]

A. P. Kattnig, J. Primot, “Model of the second-order statistic of the radiance field of natural scenes, adapted to system conceiving,” in Visual Information Processing VI, S. K. Park, R. D. Juday, eds., Proc. SPIE3074, 132–141 (1997).
[CrossRef]

Y. Goussard, G. Demoment, J. Idier, “A new algorithm for iterative deconvolution of sparse spike,” in Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1990), pp. 1547–1550.

F. Champagnat, J. Idier, “An alternative to standard maximum likelihood for Gaussian mixtures,” in Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1995), pp. 2020–2023.

E. Lehmann, Theory of Point Estimation (Wiley, New York1983).

E. D. Carvalho, D. Slock, “Maximum-likelihood blind FIR multi-channel estimation with Gaussian prior for the symbols,” in Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1997), pp. 3593–3596.

J. Seldin, R. Paxman, “Closed-loop wavefront sensing for a sparse-aperture, phased-array telescope using broadband phase diversity,” in Imaging Technology and Telescopes, J. W. Bilbro, J. B. Breckinridge, R. A. Carreras, S. R. Czyzak, M. J. Eckart, R. D. Fiete, P. S. Idell, eds., Proc. SPIE4091, 48–63 (2000).
[CrossRef]

D. A. Carrara, B. J. Thelen, R. G. Paxman, “Aberration correction of segmented-aperture telescopes by using phase diversity,” in Image Reconstruction from Incomplete Data, M. A. Fiddy, R. P. Millane, eds., Proc. SPIE4123, 56–63 (2000).
[CrossRef]

M. G. Löfdahl, A. L. Duncan, “Fast phase diversity wavefront sensor for mirror control,” in Adaptative Optical System Technologies, D. Bonaccini, R. K. Tyson, eds., Proc. SPIE3353, 952–963 (1998).
[CrossRef]

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Figures (11)

Fig. 1
Fig. 1

Phase diversity principle.

Fig. 2
Fig. 2

(a) Aberrated phase (λ/7 rms), (b) true object, (c) focused image, (d) defocused image.

Fig. 3
Fig. 3

RMSE of phase estimates as a function of noise level given in percent (it is the ratio between the noise standard deviation and the mean flux per pixel): (a) joint estimator, (b) marginal estimator. The solid, dashed and dotted curves, correspond respectively, to images of dimensions 128×128, 64×64 and 32×32 pixels. Such RMSE estimates have been obtained as empirical averages on 50 independent realizations of noise.

Fig. 4
Fig. 4

Plots of RMSE for joint phase estimates (dashed curve–right vertical axis) and joint object estimate (solid curve–left vertical axis) as a function of the value of the hyperparameter μ for an image size of 32×32 pixels. Noise level of (a) 14%, (b) 4%.

Fig. 5
Fig. 5

RMSE of joint phase estimates as a function of noise level for a near-null regularization for the three image sizes noted on graph.

Fig. 6
Fig. 6

Plots of RMSE for marginal phase estimates (dashed curve–right vertical axis) and marginal object estimate (solid curve–left vertical axis) as a function of the value of the hyperparameter μ for an image size of 32×32 pixels and a noise level of 4%.

Fig. 7
Fig. 7

Performance of marginal estimation with the true hyperparamaters (pluses) and for unsupervised estimation (diamonds) as measured by RMSE of phase estimates versus noise level.

Fig. 8
Fig. 8

RMSE of unsupervised marginal estimator (diamonds) and joint estimator with near-null regularization aberrations estimates (pluses) as a function of noise level.

Fig. 9
Fig. 9

Object restored by the joint method (left panel; near-null regularization used) and by the unsupervised marginal method (right panel). The noise level is equal to 4%.

Fig. 10
Fig. 10

Optical setup.

Fig. 11
Fig. 11

RMSE of unsupervised marginal estimator (diamonds) and the joint estimator with null regularization (pluses) aberrations estimates as a function of noise level.

Tables (1)

Tables Icon

Table 1 Values of the Coefficients Used for Simulations

Equations (40)

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

i(r)=(h*o)(r)+n(r),
h1(r)=|FT-1{P(u)exp[jϕ(u)]}|2,
h2(r)= |FT-1(P(u)exp{j[ϕ(u)+ϕd(u)]})|2,
ϕ(u)=i=4kaiZi(u).
i=Ho+n,
(oˆ, aˆ)jmap=argmaxo,a f(i1, i2, o, a; θ)=argmaxo,a f(i1|o, a; θ)f(i2|o, a; θ)×f(o; θ)f(a; θ),
f(i1, i2, o, a; θ)
=1(2π)N2/2σN2exp-12σ2 (i1-H1o)t(i1-H1o)×1(2π)N2/2σN2exp-12σ2 (i2-H2o)t(i2-H2o)×1(2π)N2/2det(Ro)1/2exp-12 (o-om)tRo-1(o-om)×1(2π)(k-3)/2det(Ra)1/2exp-12atRa-1a,
Ljmap(o, a, θ)=-ln f(i1, i2, o, a; θ)=N2ln σ2+12lndet(Ro)+12lndet(Ra)+12σ2 (i1-H1o)t(i1-H1o)+12σ2 (i2-H2o)t(i2-H2o)+12 (o-om)tRo-1(o-om)+12atRa-1a+A,
oˆ(a, θ)=(H1tH1+H2tH2+σ2Ro-1)-1(H1ti1+H2ti2+σ2Ro-1om).
Ljmap(o, a, θ)
=N2ln σ2+12vln So(v)+12lndet(Ra)+v12σ2 |ı˜1(v)-h˜1(a, v)o˜(v)|2+v12σ2 |ı˜2(v)-h˜2(a, v)o˜(v)|2+v|o˜(v)-o˜m(v)|22So(v)+12atRo-1a+A,
o˜ˆ(a, θ, v)=h˜1*(a, v)ı˜1υ+h˜2*(a, v)ı˜2υ+σ2o˜m(v)So(v)|h˜1(a, v)|2+|h˜2(a, v)|2+σ2So(v) ,
Ljmap(a, θ)=Ljmap[oˆ(a, θ), a, θ]=N2ln σ2+12vln So(v)+12v|ι˜1(v)h˜2(a, v)-ι˜2(v)h˜1(a, v)|2σ2|h˜1(a, v)|2+|h˜2(a, v)|2+σ2So(v)+12v|h˜1(a, v)o˜m(v)-ι˜1(v)|2+|h˜2(a, v)o˜m(v)-ι˜2(v)|2So(v)|h˜1(a, v)|2+|h˜2(a, v)|2+σ2So(v)+12lndet(Ra)+12atRa-1a+A.
So(v)  E[|o˜(v)-o˜m(v)|2]=k/(vop+vp)-|o˜m(v)|2,
a^map=argmaxa f(i1, i2, a; θ)=argmaxaf(i1, i2, o, a; θ)do=argmaxaf(i1|a, o; θ)f(i2|a, o; θ)×f(a; θ)f(o; θ)do.
Lmap(a, θ)=12lndet(RI)+12(I-mI)TRI-1(I-mI)+12lndet(Ra)+12atRa-1a+B,
RI=H1RoH1t+σ2IdH1RoH2tH2RoH1tH2RoH2t+σ2Id,
Lmap(a, θ)=12lndet(RI)+12v|ι˜1(v)h˜2(a, v)-ι˜2(v)h˜1(a, v)|2σ2|h˜1(a, v)|2+|h˜2(a, v)|2+σ2So(v)+12v|h˜1(a, v)o˜m(v)-ι˜1(v)|2+|h˜2(a, v)o˜m(v)-ι˜2(v)|2So(v)|h˜1(a, v)|2+|h˜2(a, v)|2+σ2So(v)+12lndet(Ra)+12atRa-1a+B.
Lmap(a, θ)=12lndet(RI)-N2ln σ2-12vln So(v)+Ljmap(a, θ)+C,
Lmap(a)=12lndet(RI)+Ljmap(a).
lndet(RI)=vln So(v)+N2ln σ2+vln|h˜1(a, v)|2+|h˜2(a, v)|2+σ2So(v).
(aˆ, θ^o, θ^n)=argmaxa,θo,θn f(i1, i2, a; θ).
Ro=F-1diag[So]F
H1=F-1diag[h˜1]F
H2=F-1diag[h˜2]F
RI-1=Q11Q12Q21Q22
Q11=[(H1RoH1t+σ2Id)-H1RoH2t(H2RoH2t+σ2Id)-1H2RoH1t]-1,
Q12=-Q11(H1RoH2t)(H2RoH2t+σ2Id)-1,
Q21=-(H2RoH2t+σ2Id)-1(H2RoH1t)Q11,
Q22=[(H2RoH2t+σ2Id)-H2RoH1t(H1RoH1t+σ2Id)-1H1RoH2t]-1.
(I-mI)TRI-1(I-mI)
=(i1-H1om)tQ11(i1-H1om)+(i1-H1om)tQ12(i2-H2om)+(i2-H2om)tQ21(i1-H1om)+(i2-H2om)tQ22(i2-H2om).
(I-mI)TRI-1(I-mI)=v|ı˜1(v)h˜2(a, v)-ı˜2(v)h˜1(a, v)|2σ2|h˜1(a, v)|2+|h˜2(a, v)|2+σ2So(v)+v|h˜1(a, v)o˜m(v)-ı˜1(v)|2+|h˜2(a, v)o˜m(v)-ı˜2(v)|2So(v)|h˜1(a, v)|2+|h˜2(a, v)|2+σ2So(v).
Δ=ABCD
det(Δ)=det(A)det(D-CA-1B).
det(RI)=det(H1RoH1t+σ2Id)det[H2RoH2t+σ2Id-H2RoH1t(H1RoH1t+σ2Id)-1H1RoH2t].
det(RI)=vσ2Id×vSo(v)×v|h˜1(a, v)|2+|h˜2(a, v)|2+σ2So(v).
argmino,a,θ Lmapalt(o, a, θ)=argmina,θargmino Lmapalt (o, a, θ)=argmina,θargmino[Ljmap(o, a, θ)]+ε(a, θ)=argmina,θ[Ljmap(a, θ)+ε(a, θ)]=argmina,θ Lmap(a, θ).

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