Abstract

Image deblurring has long been modeled as a deconvolution problem. In the literature, the point-spread function (PSF) is often assumed to be known exactly. However, in practical situations such as image acquisition in cameras, we may have incomplete knowledge of the PSF. This deblurring problem is referred to as blind deconvolution. We employ a statistical point of view of the data and use a modified maximum a posteriori approach to identify the most probable object and blur given the observed image. To facilitate computation we use an iterative method, which is an extension of the traditional expectation–maximization method, instead of direct optimization. We derive separate formulas for the updates of the estimates in each iteration to enhance the deconvolution results, which are based on the specific nature of our a priori knowledge available about the object and the blur.

© 2000 Optical Society of America

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References

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  1. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).
  2. K. R. Castleman, Digital Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1996).
  3. A. Gersho, R. Gray, Vector Quantization and Signal Compression (Kluwer Academic, Boston, Mass., 1992).
  4. E. Y. Lam, J. W. Goodman, “Discrete cosine transform domain restoration of defocused images,” Appl. Opt. 37, 6213–6218 (1998).
    [CrossRef]
  5. A. K. Katsaggelos, K.-T. Lay, “Maximum likelihood identification and restoration of images using the expectation-maximization algorithm,” in Digital Image Restoration, A. K. Katsaggelos, ed. (Springer-Verlag, New York, 1991), pp. 143–176.
  6. M. Cannon, “Blind deconvolution of spatially invariant image blurs with phase,” IEEE Trans. Acoust., Speech, Signal Process. 24, 58–63 (1976).
    [CrossRef]
  7. R. Bates, B. Quek, C. Parker, “Some implications of zero sheets for blind deconvolution and phase retrieval,” J. Opt. Soc. Am. A 7, 468–479 (1990).
    [CrossRef]
  8. D. Ghiglia, L. Romero, G. Mastin, “Systematic approach to two-dimensional blind deconvolution by zero-sheet separation,” J. Opt. Soc. Am. A 10, 1024–1036 (1993).
    [CrossRef]
  9. E. Y. Lam, J. W. Goodman, “Blind image deconvolution for symmetric blurs by polynomial factorization,” in 18th Congress of the International Commission for Optics: Optics for the Next Millennium, A. J. Glass, J. W. Goodman, M. Chang, A. H. Guenther, T. Asakura, eds., Proc. SPIE3749, 174–175 (1999).
    [CrossRef]
  10. G. Ayers, J. Dainty, “Iterative blind deconvolution method and its applications,” Opt. Lett. 13, 547–549 (1988).
    [CrossRef]
  11. Y.-L. You, M. Kaveh, “A regularization approach to joint blur identification and image restoration,” IEEE Trans. Image Process. 5, 416–428 (1996).
    [CrossRef] [PubMed]
  12. D. Kundur, D. Hatzinakos, “A novel blind deconvolution scheme for image restoration using recursive filtering,” IEEE Trans. Signal Process. 46, 375–390 (1998).
    [CrossRef]
  13. E. Y. Lam, J. W. Goodman, “Iterative blind image deconvolution in space and frequency domains,” in Sensors, Cameras, and Applications for Digital Photography, N. Sampat, T. Yeh, eds., Proc. SPIE3650, 70–77 (1999).
    [CrossRef]
  14. R. L. Lagendijk, A. M. Tekalp, J. Biemond, “Maximum likelihood image and blur identification: a unifying approach,” Opt. Eng. 29, 422–435 (1990).
    [CrossRef]
  15. K.-T. Lay, A. K. Katsaggelos, “Image identification and restoration based on the expectation-maximization algorithm,” Opt. Eng. 29, 436–446 (1990).
    [CrossRef]
  16. T. J. Hebert, K. Lu, “Expectation-maximization algorithms, null spaces, and MAP image restoration,” IEEE Trans. Image Process. 4, 1084–1095 (1995).
    [CrossRef] [PubMed]
  17. C. Bouman, K. Sauer, “A generalized Gaussian image model for edge-preserving MAP estimation,” IEEE Trans. Image Process. 2, 296–310 (1993).
    [CrossRef] [PubMed]
  18. S. Z. Li, “MAP image restoration and segmentation by constrained optimization,” IEEE Trans. Image Process. 7, 1730–1735 (1998).
    [CrossRef]
  19. L. Mugnier, J.-M. Conan, T. Fusco, V. Michau, “Joint maximum a posteriori estimation of object and PSF for turbulence degraded images,” in Bayesian Inference for Inverse Problems, A. Mohammad-Djafari, ed., Proc. SPIE3459, 50–61 (1998).
    [CrossRef]
  20. J. Markham, J.-A. Conchello, “Parametric blind deconvolution: a robust method for the simultaneous estimation of image and blur,” J. Opt. Soc. Am. A 16, 2377–2391 (1999).
    [CrossRef]
  21. G. J. McLachlan, T. Krishnan, The EM Algorithm and Extensions (Wiley, New York, 1997).
  22. A. Dempster, N. Laird, D. Rubin, “Maximum likelihood from incomplete data,” J. R. Statist. Soc. Ser. B 39, 1–38 (1977).
  23. R. J. Little, D. B. Rubin, Statistical Analysis with Missing Data (Wiley, New York, 1987).
  24. H. Andrews, B. Hunt, Digital Image Restoration (Prentice-Hall, Englewood Cliffs, N.J., 1977).
  25. A. Katsaggelos, Digital Image Restoration (Springer-Verlag, New York, 1991).
  26. A. K. Jain, Fundamentals of Digital Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1989).
  27. K. Miller, “Least-squares methods for ill-posed problems with a prescribed bound,” SIAM J. Math. Anal. 1, 52–74 (1970).
    [CrossRef]
  28. T. Kailath, Linear Systems (Prentice-Hall, Englewood Cliffs, N.J., 1980).

1999 (1)

1998 (3)

S. Z. Li, “MAP image restoration and segmentation by constrained optimization,” IEEE Trans. Image Process. 7, 1730–1735 (1998).
[CrossRef]

E. Y. Lam, J. W. Goodman, “Discrete cosine transform domain restoration of defocused images,” Appl. Opt. 37, 6213–6218 (1998).
[CrossRef]

D. Kundur, D. Hatzinakos, “A novel blind deconvolution scheme for image restoration using recursive filtering,” IEEE Trans. Signal Process. 46, 375–390 (1998).
[CrossRef]

1996 (1)

Y.-L. You, M. Kaveh, “A regularization approach to joint blur identification and image restoration,” IEEE Trans. Image Process. 5, 416–428 (1996).
[CrossRef] [PubMed]

1995 (1)

T. J. Hebert, K. Lu, “Expectation-maximization algorithms, null spaces, and MAP image restoration,” IEEE Trans. Image Process. 4, 1084–1095 (1995).
[CrossRef] [PubMed]

1993 (2)

C. Bouman, K. Sauer, “A generalized Gaussian image model for edge-preserving MAP estimation,” IEEE Trans. Image Process. 2, 296–310 (1993).
[CrossRef] [PubMed]

D. Ghiglia, L. Romero, G. Mastin, “Systematic approach to two-dimensional blind deconvolution by zero-sheet separation,” J. Opt. Soc. Am. A 10, 1024–1036 (1993).
[CrossRef]

1990 (3)

R. Bates, B. Quek, C. Parker, “Some implications of zero sheets for blind deconvolution and phase retrieval,” J. Opt. Soc. Am. A 7, 468–479 (1990).
[CrossRef]

R. L. Lagendijk, A. M. Tekalp, J. Biemond, “Maximum likelihood image and blur identification: a unifying approach,” Opt. Eng. 29, 422–435 (1990).
[CrossRef]

K.-T. Lay, A. K. Katsaggelos, “Image identification and restoration based on the expectation-maximization algorithm,” Opt. Eng. 29, 436–446 (1990).
[CrossRef]

1988 (1)

1977 (1)

A. Dempster, N. Laird, D. Rubin, “Maximum likelihood from incomplete data,” J. R. Statist. Soc. Ser. B 39, 1–38 (1977).

1976 (1)

M. Cannon, “Blind deconvolution of spatially invariant image blurs with phase,” IEEE Trans. Acoust., Speech, Signal Process. 24, 58–63 (1976).
[CrossRef]

1970 (1)

K. Miller, “Least-squares methods for ill-posed problems with a prescribed bound,” SIAM J. Math. Anal. 1, 52–74 (1970).
[CrossRef]

Andrews, H.

H. Andrews, B. Hunt, Digital Image Restoration (Prentice-Hall, Englewood Cliffs, N.J., 1977).

Ayers, G.

Bates, R.

Biemond, J.

R. L. Lagendijk, A. M. Tekalp, J. Biemond, “Maximum likelihood image and blur identification: a unifying approach,” Opt. Eng. 29, 422–435 (1990).
[CrossRef]

Bouman, C.

C. Bouman, K. Sauer, “A generalized Gaussian image model for edge-preserving MAP estimation,” IEEE Trans. Image Process. 2, 296–310 (1993).
[CrossRef] [PubMed]

Cannon, M.

M. Cannon, “Blind deconvolution of spatially invariant image blurs with phase,” IEEE Trans. Acoust., Speech, Signal Process. 24, 58–63 (1976).
[CrossRef]

Castleman, K. R.

K. R. Castleman, Digital Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1996).

Conan, J.-M.

L. Mugnier, J.-M. Conan, T. Fusco, V. Michau, “Joint maximum a posteriori estimation of object and PSF for turbulence degraded images,” in Bayesian Inference for Inverse Problems, A. Mohammad-Djafari, ed., Proc. SPIE3459, 50–61 (1998).
[CrossRef]

Conchello, J.-A.

Dainty, J.

Dempster, A.

A. Dempster, N. Laird, D. Rubin, “Maximum likelihood from incomplete data,” J. R. Statist. Soc. Ser. B 39, 1–38 (1977).

Fusco, T.

L. Mugnier, J.-M. Conan, T. Fusco, V. Michau, “Joint maximum a posteriori estimation of object and PSF for turbulence degraded images,” in Bayesian Inference for Inverse Problems, A. Mohammad-Djafari, ed., Proc. SPIE3459, 50–61 (1998).
[CrossRef]

Gersho, A.

A. Gersho, R. Gray, Vector Quantization and Signal Compression (Kluwer Academic, Boston, Mass., 1992).

Ghiglia, D.

Goodman, J. W.

E. Y. Lam, J. W. Goodman, “Discrete cosine transform domain restoration of defocused images,” Appl. Opt. 37, 6213–6218 (1998).
[CrossRef]

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

E. Y. Lam, J. W. Goodman, “Blind image deconvolution for symmetric blurs by polynomial factorization,” in 18th Congress of the International Commission for Optics: Optics for the Next Millennium, A. J. Glass, J. W. Goodman, M. Chang, A. H. Guenther, T. Asakura, eds., Proc. SPIE3749, 174–175 (1999).
[CrossRef]

E. Y. Lam, J. W. Goodman, “Iterative blind image deconvolution in space and frequency domains,” in Sensors, Cameras, and Applications for Digital Photography, N. Sampat, T. Yeh, eds., Proc. SPIE3650, 70–77 (1999).
[CrossRef]

Gray, R.

A. Gersho, R. Gray, Vector Quantization and Signal Compression (Kluwer Academic, Boston, Mass., 1992).

Hatzinakos, D.

D. Kundur, D. Hatzinakos, “A novel blind deconvolution scheme for image restoration using recursive filtering,” IEEE Trans. Signal Process. 46, 375–390 (1998).
[CrossRef]

Hebert, T. J.

T. J. Hebert, K. Lu, “Expectation-maximization algorithms, null spaces, and MAP image restoration,” IEEE Trans. Image Process. 4, 1084–1095 (1995).
[CrossRef] [PubMed]

Hunt, B.

H. Andrews, B. Hunt, Digital Image Restoration (Prentice-Hall, Englewood Cliffs, N.J., 1977).

Jain, A. K.

A. K. Jain, Fundamentals of Digital Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1989).

Kailath, T.

T. Kailath, Linear Systems (Prentice-Hall, Englewood Cliffs, N.J., 1980).

Katsaggelos, A.

A. Katsaggelos, Digital Image Restoration (Springer-Verlag, New York, 1991).

Katsaggelos, A. K.

K.-T. Lay, A. K. Katsaggelos, “Image identification and restoration based on the expectation-maximization algorithm,” Opt. Eng. 29, 436–446 (1990).
[CrossRef]

A. K. Katsaggelos, K.-T. Lay, “Maximum likelihood identification and restoration of images using the expectation-maximization algorithm,” in Digital Image Restoration, A. K. Katsaggelos, ed. (Springer-Verlag, New York, 1991), pp. 143–176.

Kaveh, M.

Y.-L. You, M. Kaveh, “A regularization approach to joint blur identification and image restoration,” IEEE Trans. Image Process. 5, 416–428 (1996).
[CrossRef] [PubMed]

Krishnan, T.

G. J. McLachlan, T. Krishnan, The EM Algorithm and Extensions (Wiley, New York, 1997).

Kundur, D.

D. Kundur, D. Hatzinakos, “A novel blind deconvolution scheme for image restoration using recursive filtering,” IEEE Trans. Signal Process. 46, 375–390 (1998).
[CrossRef]

Lagendijk, R. L.

R. L. Lagendijk, A. M. Tekalp, J. Biemond, “Maximum likelihood image and blur identification: a unifying approach,” Opt. Eng. 29, 422–435 (1990).
[CrossRef]

Laird, N.

A. Dempster, N. Laird, D. Rubin, “Maximum likelihood from incomplete data,” J. R. Statist. Soc. Ser. B 39, 1–38 (1977).

Lam, E. Y.

E. Y. Lam, J. W. Goodman, “Discrete cosine transform domain restoration of defocused images,” Appl. Opt. 37, 6213–6218 (1998).
[CrossRef]

E. Y. Lam, J. W. Goodman, “Blind image deconvolution for symmetric blurs by polynomial factorization,” in 18th Congress of the International Commission for Optics: Optics for the Next Millennium, A. J. Glass, J. W. Goodman, M. Chang, A. H. Guenther, T. Asakura, eds., Proc. SPIE3749, 174–175 (1999).
[CrossRef]

E. Y. Lam, J. W. Goodman, “Iterative blind image deconvolution in space and frequency domains,” in Sensors, Cameras, and Applications for Digital Photography, N. Sampat, T. Yeh, eds., Proc. SPIE3650, 70–77 (1999).
[CrossRef]

Lay, K.-T.

K.-T. Lay, A. K. Katsaggelos, “Image identification and restoration based on the expectation-maximization algorithm,” Opt. Eng. 29, 436–446 (1990).
[CrossRef]

A. K. Katsaggelos, K.-T. Lay, “Maximum likelihood identification and restoration of images using the expectation-maximization algorithm,” in Digital Image Restoration, A. K. Katsaggelos, ed. (Springer-Verlag, New York, 1991), pp. 143–176.

Li, S. Z.

S. Z. Li, “MAP image restoration and segmentation by constrained optimization,” IEEE Trans. Image Process. 7, 1730–1735 (1998).
[CrossRef]

Little, R. J.

R. J. Little, D. B. Rubin, Statistical Analysis with Missing Data (Wiley, New York, 1987).

Lu, K.

T. J. Hebert, K. Lu, “Expectation-maximization algorithms, null spaces, and MAP image restoration,” IEEE Trans. Image Process. 4, 1084–1095 (1995).
[CrossRef] [PubMed]

Markham, J.

Mastin, G.

McLachlan, G. J.

G. J. McLachlan, T. Krishnan, The EM Algorithm and Extensions (Wiley, New York, 1997).

Michau, V.

L. Mugnier, J.-M. Conan, T. Fusco, V. Michau, “Joint maximum a posteriori estimation of object and PSF for turbulence degraded images,” in Bayesian Inference for Inverse Problems, A. Mohammad-Djafari, ed., Proc. SPIE3459, 50–61 (1998).
[CrossRef]

Miller, K.

K. Miller, “Least-squares methods for ill-posed problems with a prescribed bound,” SIAM J. Math. Anal. 1, 52–74 (1970).
[CrossRef]

Mugnier, L.

L. Mugnier, J.-M. Conan, T. Fusco, V. Michau, “Joint maximum a posteriori estimation of object and PSF for turbulence degraded images,” in Bayesian Inference for Inverse Problems, A. Mohammad-Djafari, ed., Proc. SPIE3459, 50–61 (1998).
[CrossRef]

Parker, C.

Quek, B.

Romero, L.

Rubin, D.

A. Dempster, N. Laird, D. Rubin, “Maximum likelihood from incomplete data,” J. R. Statist. Soc. Ser. B 39, 1–38 (1977).

Rubin, D. B.

R. J. Little, D. B. Rubin, Statistical Analysis with Missing Data (Wiley, New York, 1987).

Sauer, K.

C. Bouman, K. Sauer, “A generalized Gaussian image model for edge-preserving MAP estimation,” IEEE Trans. Image Process. 2, 296–310 (1993).
[CrossRef] [PubMed]

Tekalp, A. M.

R. L. Lagendijk, A. M. Tekalp, J. Biemond, “Maximum likelihood image and blur identification: a unifying approach,” Opt. Eng. 29, 422–435 (1990).
[CrossRef]

You, Y.-L.

Y.-L. You, M. Kaveh, “A regularization approach to joint blur identification and image restoration,” IEEE Trans. Image Process. 5, 416–428 (1996).
[CrossRef] [PubMed]

Appl. Opt. (1)

IEEE Trans. Acoust., Speech, Signal Process. (1)

M. Cannon, “Blind deconvolution of spatially invariant image blurs with phase,” IEEE Trans. Acoust., Speech, Signal Process. 24, 58–63 (1976).
[CrossRef]

IEEE Trans. Image Process. (4)

Y.-L. You, M. Kaveh, “A regularization approach to joint blur identification and image restoration,” IEEE Trans. Image Process. 5, 416–428 (1996).
[CrossRef] [PubMed]

T. J. Hebert, K. Lu, “Expectation-maximization algorithms, null spaces, and MAP image restoration,” IEEE Trans. Image Process. 4, 1084–1095 (1995).
[CrossRef] [PubMed]

C. Bouman, K. Sauer, “A generalized Gaussian image model for edge-preserving MAP estimation,” IEEE Trans. Image Process. 2, 296–310 (1993).
[CrossRef] [PubMed]

S. Z. Li, “MAP image restoration and segmentation by constrained optimization,” IEEE Trans. Image Process. 7, 1730–1735 (1998).
[CrossRef]

IEEE Trans. Signal Process. (1)

D. Kundur, D. Hatzinakos, “A novel blind deconvolution scheme for image restoration using recursive filtering,” IEEE Trans. Signal Process. 46, 375–390 (1998).
[CrossRef]

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

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

A. Dempster, N. Laird, D. Rubin, “Maximum likelihood from incomplete data,” J. R. Statist. Soc. Ser. B 39, 1–38 (1977).

Opt. Eng. (2)

R. L. Lagendijk, A. M. Tekalp, J. Biemond, “Maximum likelihood image and blur identification: a unifying approach,” Opt. Eng. 29, 422–435 (1990).
[CrossRef]

K.-T. Lay, A. K. Katsaggelos, “Image identification and restoration based on the expectation-maximization algorithm,” Opt. Eng. 29, 436–446 (1990).
[CrossRef]

Opt. Lett. (1)

SIAM J. Math. Anal. (1)

K. Miller, “Least-squares methods for ill-posed problems with a prescribed bound,” SIAM J. Math. Anal. 1, 52–74 (1970).
[CrossRef]

Other (13)

T. Kailath, Linear Systems (Prentice-Hall, Englewood Cliffs, N.J., 1980).

G. J. McLachlan, T. Krishnan, The EM Algorithm and Extensions (Wiley, New York, 1997).

R. J. Little, D. B. Rubin, Statistical Analysis with Missing Data (Wiley, New York, 1987).

H. Andrews, B. Hunt, Digital Image Restoration (Prentice-Hall, Englewood Cliffs, N.J., 1977).

A. Katsaggelos, Digital Image Restoration (Springer-Verlag, New York, 1991).

A. K. Jain, Fundamentals of Digital Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1989).

E. Y. Lam, J. W. Goodman, “Iterative blind image deconvolution in space and frequency domains,” in Sensors, Cameras, and Applications for Digital Photography, N. Sampat, T. Yeh, eds., Proc. SPIE3650, 70–77 (1999).
[CrossRef]

L. Mugnier, J.-M. Conan, T. Fusco, V. Michau, “Joint maximum a posteriori estimation of object and PSF for turbulence degraded images,” in Bayesian Inference for Inverse Problems, A. Mohammad-Djafari, ed., Proc. SPIE3459, 50–61 (1998).
[CrossRef]

E. Y. Lam, J. W. Goodman, “Blind image deconvolution for symmetric blurs by polynomial factorization,” in 18th Congress of the International Commission for Optics: Optics for the Next Millennium, A. J. Glass, J. W. Goodman, M. Chang, A. H. Guenther, T. Asakura, eds., Proc. SPIE3749, 174–175 (1999).
[CrossRef]

A. K. Katsaggelos, K.-T. Lay, “Maximum likelihood identification and restoration of images using the expectation-maximization algorithm,” in Digital Image Restoration, A. K. Katsaggelos, ed. (Springer-Verlag, New York, 1991), pp. 143–176.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

K. R. Castleman, Digital Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1996).

A. Gersho, R. Gray, Vector Quantization and Signal Compression (Kluwer Academic, Boston, Mass., 1992).

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Figures (5)

Fig. 1
Fig. 1

“Baboon” test image.

Fig. 2
Fig. 2

Simulated image restoration: (a) blurred image, (b) image restored by our first method, (c) image restored by our second method, (d) image restored by the ML method.

Fig. 3
Fig. 3

Blur estimation error.

Fig. 4
Fig. 4

Simulated image restoration for uniform noise: (a) blurred image, (b) image restored by our first method.

Fig. 5
Fig. 5

Image restoration for real images: (a) blurred image, (b) image restored by our first method.

Equations (50)

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

Gi(fx, fy)=H(fx, fy)Gg(fx, fy),
i(x, y)=h(x, y)*g(x, y)+n(x, y),
p(Θ|y)=p(y|Θ)p(Θ)p(y)
p(y|Θ)p(Θ),
Θˆ=arg maxΘΩθ p(Θ|y).
=arg maxΘΩθ[log p(y|Θ)+log p(Θ)].
Estep:L(Θ)=E[log p(z; Θ)|y; Θ^(k)],
Mstep:Θ^(k+1)=arg maxΘΩθ[L(Θ)+log p(Θ)],
pg(g)=1[det(2πϕg)]1/2 exp-12 g Tϕg-1g,
Θˆ=arg maxΘΩθ p(Θ|i)
=arg maxΘΩθ[log p(i|Θ)+log p(Θ)],
ϕz=ϕgϕgiϕigϕi
=ϕgϕgHTHϕgHϕgHT+ϕn.
log pz(z)=-12 [log (2π)+log det(ϕz)+zTϕz-1z].
Estep:L(Θ)=E[log p(z; Θ)|i; Θ^(k)],
Mstep:Θ^(k+1)=arg maxΘΩθ[L(Θ)+log p(Θ)].
E[log pz(z)|i]=-12 {log(2π)+log det(ϕz)+E[gTϕg-1g|i]+E[nTϕn-1n|i]}.
log det(ϕz)=fxfy log Φg(fx, fy)+fxfy log Φn(fx, fy),
Tr{ϕg-1E[ggT|i]}=Tr{ϕg-1ϕg|i}+Tr{ϕg-1(E[g|i]E[g|i]T)}.
Wg | i(k)(fx, fy)=H*(k)(fx, fy)|H(k)(fx, fy)|2+Φn(fx, fy)Φg(k)(fx, fy) Gi(fx, fy).
Φg | i(k)(fx, fy)=Φg(k)(fx, fy)Φi(k)(fx, fy) Φn(fx, fy).
E[gTϕg-1g|i]
=fxfy Φg|i(k)(fx, fy)+(1/N2)|Wg|i(k)(fx, fy)|2Φg(fx, fy).
L(Θ)=K-12 fxfylog Φg(fx, fy)+Φg|i(k)(fx, fy)+(1/N2)|Wg|i(k)(fx, fy)|2Φg(fx, fy).
L˜(Θ)=fxfylog Φg(fx, fy)+Φg|i(k)(fx, fy)+(1/N2)|Wg|i(k)(fx, fy)|2Φg(fx, fy)+αV(fx, fy)Φg(fx, fy).
Φg(k+1)(fx, fy)=-1+{1+4αV(fx, fy)[Φg|i(k)(fx, fy)+(1/N2)|Wg|i(k)(fx, fy)|2]}1/22αV(fα, fy).
Φg(k+1)(fx, fy)=-1+1+12 {4αV(fx, fy)[Φg|i(k)(fx, fy)+(1/N2)|Wg|i(k)(fx, fy)|2]}2αV(fx, fy)
=Φg|i(k)(fx, fy)+1N2 |Wg|i(k)(fx, fy)|2,
Φg(k+1)(fx, fy)
=1N2 fxfy[Φg|i(k)(fx, fy)+1N2 |Wg|i(k)(fx, fy)|2].
H(fx, fy)Gi(fx, fy)Wg|i(fx, fy)1N2 Gi(fx, fy)Wg|i*(fx, fy)Φg(fx, fy).
minimizefxfyH(fx, fy)
-Gi(fx, fy)Wg|i*(k)(fx, fy)N2Φg(k+1)(fx, fy)2
subjecttoHCH.
SNR(g, n)=10 log10xy[g(x, y)]2xy[n(x, y)]2,
v=-1-1-1-18-1-1-1-1.
ϕz-1=ϕg-1+HTϕn-1H-HTϕn-1-ϕn-1Hϕn-1.
zTϕz-1z=[gTiT]ϕg-1+HTϕn-1H-HTϕn-1-ϕn-1Hϕn-1 gi
=gT(ϕg-1+HTϕn-1H)g+gT(-HTϕn-1)i+iT(-ϕn-1H)g+iTϕn-1i
=gTϕg-1g+(iT-nT)ϕn-1(i-n)-(iT-nT)ϕn-1i-iTϕn-1(i-n)+iTϕn-1i
=gTϕg-1g+nTϕn-1n.
log det(ϕz)=log det(HϕgϕgHT+ϕgϕn-HϕgϕgHT)
=log det(ϕgϕn)
=log det(ϕg)+log det(ϕn).
ϕg|i=ϕg-ϕgiϕi-1ϕig
=ϕg-ϕgHϕi-1Hϕg.
Φg|i(k)(fx, fy)
=Φg(k)(fx, fy)-|H(k)(fx, fy)|2Φg(k)(fx, fy)Φi(k)(fx, fy) Φg(k)(fx, fy)
=Φi(k)(fx, fy)-|H(k)(fx, fy)|2Φg(k)(fx, fy)Φi(k)(fx, fy)×Φg(k)(fx, fy)
=Φg(k)(fx, fy)Φi(k)(fx, fy) Φn(fx, fy).

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