Abstract

This paper tackles the problem of image deconvolution with joint estimation of point spread function (PSF) parameters and hyperparameters. Within a Bayesian framework, the solution is inferred via a global a posteriori law for unknown parameters and object. The estimate is chosen as the posterior mean, numerically calculated by means of a Monte Carlo Markov chain algorithm. The estimates are efficiently computed in the Fourier domain, and the effectiveness of the method is shown on simulated examples. Results show precise estimates for PSF parameters and hyperparameters as well as precise image estimates including restoration of high frequencies and spatial details, within a global and coherent approach.

© 2010 Optical Society of America

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2009 (4)

N. Dobigeon, A. Hero, and J.-Y. Tourneret, “Hierarchical Bayesian sparse image reconstruction with application to MRFM,” IEEE Trans. Image Process. (2009).

F. Chen and J. Ma, “An empirical identification method of Gaussian blur parameter for image deblurring,” IEEE Trans. Signal Process. (2009).

Z. Xu and E. Y. Lam, “Maximum a posteriori blind image deconvolution with Huber–Markov random-field regularization,” Opt. Lett. 34, 1453–1455 (2009).
[CrossRef] [PubMed]

P. Pankajakshani, B. Zhang, L. Blanc-Féraud, Z. Kam, J.-C. Olivo-Marin, and J. Zerubia, “Blind deconvolution for thin-layered confocal imaging,” Appl. Opt. 48, 4437–4448 (2009).
[CrossRef]

2008 (5)

T. Bishop, R. Molina, and J. Hopgood, “Blind restoration of blurred photographs via AR modelling and MCMC,” in Proceedings of 15th IEEE International Conference on Image Processing, 2008, ICIP 2008 (IEEE Signal Processing Society, 2008).
[CrossRef]

J.Idier, ed., Bayesian Approach to Inverse Problems (Wiley, 2008).
[CrossRef]

T. Rodet, F. Orieux, J.-F. Giovannelli, and A. Abergel, “Data inversion for over-resolved spectral imaging in astronomy,” IEEE J. Sel. Top. Signal Process. 2, 802–811 (2008).
[CrossRef]

J.-F. Giovannelli, “Unsupervised Bayesian convex deconvolution based on a field with an explicit partition function,” IEEE Trans. Image Process. 17, 16–26 (2008).
[CrossRef] [PubMed]

E. Thiébaut, “MiRA: an effective imaging algorithm for optical interferometry,” Proc. SPIE 7013, 70131-I (2008).
[CrossRef]

2007 (2)

2006 (1)

R. Molina, J. Mateos, and A. K. Katsaggelos, “Blind deconvolution using a variational approach to parameter, image, and blur estimation,” IEEE Trans. Image Process. 15, 3715–3727 (2006).
[CrossRef] [PubMed]

2004 (2)

2003 (2)

D. MacKay, Information Theory, Inference, and Learning Algorithms (Cambridge Univ. Press, 2003).

E. T. Jaynes, Probability Theory: The Logic of Science (Cambridge Univ. Press, 2003).
[CrossRef]

2002 (2)

A. Jalobeanu, L. Blanc-Féraud, and J. Zerubia, “Hyperparameter estimation for satellite image restoration by a MCMC maximum likelihood method,” Pattern Recogn. 35, 341–352 (2002).
[CrossRef]

A. Jalobeanu, L. Blanc-Feraud, and J. Zerubia, “Estimation of blur and noise parameters in remote sensing,” in Proceedings of 2002 IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP 2002) (IEEE Signal Processing Society, 2002), Vol. 4, pp. 3580–3583.

2001 (1)

2000 (2)

E. Y. Lam and J. W. Goodman, “Iterative statistical approach to blind image deconvolution,” J. Opt. Soc. Am. A 17, 1177–1184 (2000).
[CrossRef]

C. P. Robert and G. Casella, Monte-Carlo Statistical Methods, Springer Texts in Statistics (Springer, 2000).

1999 (3)

P. Brémaud, Markov Chains. Gibbs Fields, Monte Carlo Simulation, and Queues, Texts in Applied Mathematics 31 (Springer, 1999).

T. Fusco, J.-P. Véran, J.-M. Conan, and L. M. Mugnier, “Myopic deconvolution method for adaptive optics images of stellar fields,” Astron. Astrophys. Suppl. Ser. 134, 193 (1999).
[CrossRef]

X. Descombes, R. Morris, J. Zerubia, and M. Berthod, “Estimation of Markov random field prior parameters using Markov chain Monte Carlo maximum likelihood,” IEEE Trans. Image Process. 8, 954–963 (1999).
[CrossRef]

1998 (1)

1997 (2)

P. Charbonnier, L. Blanc-Féraud, G. Aubert, and M. Barlaud, “Deterministic edge-preserving regularization in computed imaging,” IEEE Trans. Image Process. 6, 298–311 (1997).
[CrossRef] [PubMed]

M. Calder and R. A. Davis, “Introduction to Whittle (1953) ‘The analysis of multiple stationary time series’,” Breakthroughs in Statistics 3, 141–148 (1997).
[CrossRef]

1996 (1)

R. E. Kass and L. Wasserman, “The selection of prior distributions by formal rules,” J. Am. Stat. Assoc. 91, 1343–1370 (1996).
[CrossRef]

1995 (3)

D. Geman and C. Yang, “Nonlinear image recovery with half-quadratic regularization,” IEEE Trans. Image Process. 4, 932–946 (1995).
[CrossRef] [PubMed]

J. A. O’Sullivan, “Roughness penalties on finite domains,” IEEE Trans. Image Process. 4, 1258–1268 (1995).
[CrossRef] [PubMed]

E. Thiébaut and J.-M. Conan, “Strict a priori constraints for maximum likelihood blind deconvolution,” J. Opt. Soc. Am. A 12, 485–492 (1995).
[CrossRef]

1994 (1)

H. R. Künsch, “Robust priors for smoothing and image restoration,” Ann. Inst. Stat. Math. 46, 1–19 (1994).
[CrossRef]

1993 (2)

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

S. Lang, Real and Functional Analysis (Springer, 1993).
[CrossRef]

1992 (1)

K. Mardia, J. Kent, and J. Bibby, Multivariate Analysis (Academic, 1992), Chap. 2, pp. 36–43.

1991 (1)

P. J. Brockwell and R. A. Davis, Time Series: Theory and Methods (Springer-Verlag, 1991).
[CrossRef]

1989 (1)

G. Demoment, “Image reconstruction and restoration: overview of common estimation structure and problems,” IEEE Trans. Acoust., Speech, Signal Process. ASSP-37, 2024–2036 (1989).
[CrossRef]

1984 (1)

S. Geman and D. Geman, “Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images,” IEEE Trans. Pattern Anal. Mach. Intell. 6, 721–741 (1984).
[CrossRef]

1977 (1)

A. Tikhonov and V. Arsenin, Solutions of Ill-Posed Problems (Winston, 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]

1972 (2)

B. R. Hunt, “Deconvolution of linear systems by constrained regression and its relationship to the Wiener theory,” IEEE Trans. Autom. Control AC-17, 703–705 (1972).
[CrossRef]

G. E. P. Box and G. C. Tiao, Bayesian Inference in Statistical Analysis (Addison-Wesley, 1972).

1971 (1)

B. R. Hunt, “A matrix theory proof of the discrete convolution theorem,” IEEE Trans. Autom. Control AC-19, 285–288 (1971).

1962 (1)

S. Twomey, “On the numerical solution of Fredholm integral equations of the first kind by the inversion of the linear system produced by quadrature,” J. Assoc. Comput. Mach. 10, 97–101 (1962).
[CrossRef]

Abergel, A.

T. Rodet, F. Orieux, J.-F. Giovannelli, and A. Abergel, “Data inversion for over-resolved spectral imaging in astronomy,” IEEE J. Sel. Top. Signal Process. 2, 802–811 (2008).
[CrossRef]

Arsenin, V.

A. Tikhonov and V. Arsenin, Solutions of Ill-Posed Problems (Winston, 1977).

Aubert, G.

P. Charbonnier, L. Blanc-Féraud, G. Aubert, and M. Barlaud, “Deterministic edge-preserving regularization in computed imaging,” IEEE Trans. Image Process. 6, 298–311 (1997).
[CrossRef] [PubMed]

Barlaud, M.

P. Charbonnier, L. Blanc-Féraud, G. Aubert, and M. Barlaud, “Deterministic edge-preserving regularization in computed imaging,” IEEE Trans. Image Process. 6, 298–311 (1997).
[CrossRef] [PubMed]

Berthod, M.

X. Descombes, R. Morris, J. Zerubia, and M. Berthod, “Estimation of Markov random field prior parameters using Markov chain Monte Carlo maximum likelihood,” IEEE Trans. Image Process. 8, 954–963 (1999).
[CrossRef]

Bibby, J.

K. Mardia, J. Kent, and J. Bibby, Multivariate Analysis (Academic, 1992), Chap. 2, pp. 36–43.

Bishop, T.

T. Bishop, R. Molina, and J. Hopgood, “Blind restoration of blurred photographs via AR modelling and MCMC,” in Proceedings of 15th IEEE International Conference on Image Processing, 2008, ICIP 2008 (IEEE Signal Processing Society, 2008).
[CrossRef]

Blanc-Feraud, L.

A. Jalobeanu, L. Blanc-Feraud, and J. Zerubia, “Estimation of blur and noise parameters in remote sensing,” in Proceedings of 2002 IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP 2002) (IEEE Signal Processing Society, 2002), Vol. 4, pp. 3580–3583.

Blanc-Féraud, L.

P. Pankajakshani, B. Zhang, L. Blanc-Féraud, Z. Kam, J.-C. Olivo-Marin, and J. Zerubia, “Blind deconvolution for thin-layered confocal imaging,” Appl. Opt. 48, 4437–4448 (2009).
[CrossRef]

A. Jalobeanu, L. Blanc-Féraud, and J. Zerubia, “Hyperparameter estimation for satellite image restoration by a MCMC maximum likelihood method,” Pattern Recogn. 35, 341–352 (2002).
[CrossRef]

P. Charbonnier, L. Blanc-Féraud, G. Aubert, and M. Barlaud, “Deterministic edge-preserving regularization in computed imaging,” IEEE Trans. Image Process. 6, 298–311 (1997).
[CrossRef] [PubMed]

Bouman, C. A.

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

Box, G. E. P.

G. E. P. Box and G. C. Tiao, Bayesian Inference in Statistical Analysis (Addison-Wesley, 1972).

Brémaud, P.

P. Brémaud, Markov Chains. Gibbs Fields, Monte Carlo Simulation, and Queues, Texts in Applied Mathematics 31 (Springer, 1999).

Brockwell, P. J.

P. J. Brockwell and R. A. Davis, Time Series: Theory and Methods (Springer-Verlag, 1991).
[CrossRef]

Calder, M.

M. Calder and R. A. Davis, “Introduction to Whittle (1953) ‘The analysis of multiple stationary time series’,” Breakthroughs in Statistics 3, 141–148 (1997).
[CrossRef]

Cannon, M.

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

Casella, G.

C. P. Robert and G. Casella, Monte-Carlo Statistical Methods, Springer Texts in Statistics (Springer, 2000).

Charbonnier, P.

P. Charbonnier, L. Blanc-Féraud, G. Aubert, and M. Barlaud, “Deterministic edge-preserving regularization in computed imaging,” IEEE Trans. Image Process. 6, 298–311 (1997).
[CrossRef] [PubMed]

Chavel, P.

Chen, F.

F. Chen and J. Ma, “An empirical identification method of Gaussian blur parameter for image deblurring,” IEEE Trans. Signal Process. (2009).

Conan, J.-M.

Davis, R. A.

M. Calder and R. A. Davis, “Introduction to Whittle (1953) ‘The analysis of multiple stationary time series’,” Breakthroughs in Statistics 3, 141–148 (1997).
[CrossRef]

P. J. Brockwell and R. A. Davis, Time Series: Theory and Methods (Springer-Verlag, 1991).
[CrossRef]

Demoment, G.

G. Demoment, “Image reconstruction and restoration: overview of common estimation structure and problems,” IEEE Trans. Acoust., Speech, Signal Process. ASSP-37, 2024–2036 (1989).
[CrossRef]

Descombes, X.

X. Descombes, R. Morris, J. Zerubia, and M. Berthod, “Estimation of Markov random field prior parameters using Markov chain Monte Carlo maximum likelihood,” IEEE Trans. Image Process. 8, 954–963 (1999).
[CrossRef]

Dobigeon, N.

N. Dobigeon, A. Hero, and J.-Y. Tourneret, “Hierarchical Bayesian sparse image reconstruction with application to MRFM,” IEEE Trans. Image Process. (2009).

Fusco, T.

Galatsanos, N. P.

A. C. Likas and N. P. Galatsanos, “A variational approach for Bayesian blind image deconvolution,” IEEE Trans. Image Process. 52, 2222–2233 (2004).

Geman, D.

D. Geman and C. Yang, “Nonlinear image recovery with half-quadratic regularization,” IEEE Trans. Image Process. 4, 932–946 (1995).
[CrossRef] [PubMed]

S. Geman and D. Geman, “Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images,” IEEE Trans. Pattern Anal. Mach. Intell. 6, 721–741 (1984).
[CrossRef]

Geman, S.

S. Geman and D. Geman, “Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images,” IEEE Trans. Pattern Anal. Mach. Intell. 6, 721–741 (1984).
[CrossRef]

Giovannelli, J.-F.

J.-F. Giovannelli, “Unsupervised Bayesian convex deconvolution based on a field with an explicit partition function,” IEEE Trans. Image Process. 17, 16–26 (2008).
[CrossRef] [PubMed]

T. Rodet, F. Orieux, J.-F. Giovannelli, and A. Abergel, “Data inversion for over-resolved spectral imaging in astronomy,” IEEE J. Sel. Top. Signal Process. 2, 802–811 (2008).
[CrossRef]

Goodman, J. W.

Hero, A.

N. Dobigeon, A. Hero, and J.-Y. Tourneret, “Hierarchical Bayesian sparse image reconstruction with application to MRFM,” IEEE Trans. Image Process. (2009).

Hopgood, J.

T. Bishop, R. Molina, and J. Hopgood, “Blind restoration of blurred photographs via AR modelling and MCMC,” in Proceedings of 15th IEEE International Conference on Image Processing, 2008, ICIP 2008 (IEEE Signal Processing Society, 2008).
[CrossRef]

Hunt, B. R.

B. R. Hunt, “Deconvolution of linear systems by constrained regression and its relationship to the Wiener theory,” IEEE Trans. Autom. Control AC-17, 703–705 (1972).
[CrossRef]

B. R. Hunt, “A matrix theory proof of the discrete convolution theorem,” IEEE Trans. Autom. Control AC-19, 285–288 (1971).

Jalobeanu, A.

A. Jalobeanu, L. Blanc-Féraud, and J. Zerubia, “Hyperparameter estimation for satellite image restoration by a MCMC maximum likelihood method,” Pattern Recogn. 35, 341–352 (2002).
[CrossRef]

A. Jalobeanu, L. Blanc-Feraud, and J. Zerubia, “Estimation of blur and noise parameters in remote sensing,” in Proceedings of 2002 IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP 2002) (IEEE Signal Processing Society, 2002), Vol. 4, pp. 3580–3583.

Jaynes, E. T.

E. T. Jaynes, Probability Theory: The Logic of Science (Cambridge Univ. Press, 2003).
[CrossRef]

Kam, Z.

Kass, R. E.

R. E. Kass and L. Wasserman, “The selection of prior distributions by formal rules,” J. Am. Stat. Assoc. 91, 1343–1370 (1996).
[CrossRef]

Katsaggelos, A. K.

R. Molina, J. Mateos, and A. K. Katsaggelos, “Blind deconvolution using a variational approach to parameter, image, and blur estimation,” IEEE Trans. Image Process. 15, 3715–3727 (2006).
[CrossRef] [PubMed]

Kent, J.

K. Mardia, J. Kent, and J. Bibby, Multivariate Analysis (Academic, 1992), Chap. 2, pp. 36–43.

Künsch, H. R.

H. R. Künsch, “Robust priors for smoothing and image restoration,” Ann. Inst. Stat. Math. 46, 1–19 (1994).
[CrossRef]

Lalanne, P.

Lam, E. Y.

Lang, S.

S. Lang, Real and Functional Analysis (Springer, 1993).
[CrossRef]

Likas, A. C.

A. C. Likas and N. P. Galatsanos, “A variational approach for Bayesian blind image deconvolution,” IEEE Trans. Image Process. 52, 2222–2233 (2004).

Ma, J.

F. Chen and J. Ma, “An empirical identification method of Gaussian blur parameter for image deblurring,” IEEE Trans. Signal Process. (2009).

MacKay, D.

D. MacKay, Information Theory, Inference, and Learning Algorithms (Cambridge Univ. Press, 2003).

Mardia, K.

K. Mardia, J. Kent, and J. Bibby, Multivariate Analysis (Academic, 1992), Chap. 2, pp. 36–43.

Mateos, J.

R. Molina, J. Mateos, and A. K. Katsaggelos, “Blind deconvolution using a variational approach to parameter, image, and blur estimation,” IEEE Trans. Image Process. 15, 3715–3727 (2006).
[CrossRef] [PubMed]

Michau, V.

Molina, R.

T. Bishop, R. Molina, and J. Hopgood, “Blind restoration of blurred photographs via AR modelling and MCMC,” in Proceedings of 15th IEEE International Conference on Image Processing, 2008, ICIP 2008 (IEEE Signal Processing Society, 2008).
[CrossRef]

R. Molina, J. Mateos, and A. K. Katsaggelos, “Blind deconvolution using a variational approach to parameter, image, and blur estimation,” IEEE Trans. Image Process. 15, 3715–3727 (2006).
[CrossRef] [PubMed]

Morris, R.

X. Descombes, R. Morris, J. Zerubia, and M. Berthod, “Estimation of Markov random field prior parameters using Markov chain Monte Carlo maximum likelihood,” IEEE Trans. Image Process. 8, 954–963 (1999).
[CrossRef]

Mugnier, L.

Mugnier, L. M.

T. Fusco, J.-P. Véran, J.-M. Conan, and L. M. Mugnier, “Myopic deconvolution method for adaptive optics images of stellar fields,” Astron. Astrophys. Suppl. Ser. 134, 193 (1999).
[CrossRef]

O’Sullivan, J. A.

J. A. O’Sullivan, “Roughness penalties on finite domains,” IEEE Trans. Image Process. 4, 1258–1268 (1995).
[CrossRef] [PubMed]

Olivo-Marin, J.-C.

Orieux, F.

T. Rodet, F. Orieux, J.-F. Giovannelli, and A. Abergel, “Data inversion for over-resolved spectral imaging in astronomy,” IEEE J. Sel. Top. Signal Process. 2, 802–811 (2008).
[CrossRef]

Pankajakshani, P.

Prévost, D.

Robert, C. P.

C. P. Robert and G. Casella, Monte-Carlo Statistical Methods, Springer Texts in Statistics (Springer, 2000).

Rodet, T.

T. Rodet, F. Orieux, J.-F. Giovannelli, and A. Abergel, “Data inversion for over-resolved spectral imaging in astronomy,” IEEE J. Sel. Top. Signal Process. 2, 802–811 (2008).
[CrossRef]

Rousset, G.

Sauer, K. D.

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

Thiébaut, E.

E. Thiébaut, “MiRA: an effective imaging algorithm for optical interferometry,” Proc. SPIE 7013, 70131-I (2008).
[CrossRef]

E. Thiébaut and J.-M. Conan, “Strict a priori constraints for maximum likelihood blind deconvolution,” J. Opt. Soc. Am. A 12, 485–492 (1995).
[CrossRef]

Tiao, G. C.

G. E. P. Box and G. C. Tiao, Bayesian Inference in Statistical Analysis (Addison-Wesley, 1972).

Tikhonov, A.

A. Tikhonov and V. Arsenin, Solutions of Ill-Posed Problems (Winston, 1977).

Tourneret, J.-Y.

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Figures (8)

Fig. 1
Fig. 1

(a) 128 × 128 sample of the a priori law for the object with γ 0 = 1 and γ 1 = 2 . (b) Data computed with the PSF shown in Fig. 2. (c) and (d) Estimates with non-myopic and the myopic estimate, respectively. Profiles correspond to the 68th line of the image.

Fig. 2
Fig. 2

PSF with w α = 20 , w β = 7 and φ = π 3 . The x axis and y axis are reduced frequency.

Fig. 3
Fig. 3

Circular average of the empirical power spectral density of the image, the convolved image, the data (convolved image corrupted by noise), and the estimates, in radial frequency with the y axis in logarithmic scale. The x axis is the radial frequency.

Fig. 4
Fig. 4

Computation of the best parameters in the sense e, Eq. (46). The symbol × is the minimum and the dot is the estimated value with our approach. The y axes of γ ϵ and γ 1 are in logarithmic scale.

Fig. 5
Fig. 5

Histograms and chains for the non-myopic case [(a) and (c)] and the myopic case [(b) and (d)] for γ ϵ and γ 1 , respectively. The symbol × on the y axes localizes the initial value, and the dashed line corresponds to the true value. The x axes are the iteration’s index for the chains (bottom of figures) and the parameter value for the histograms (top of figures).

Fig. 6
Fig. 6

Histogram and chain for the PSF parameters (a) w α , (b) w β , and (c) φ. The symbol × on the y axes localizes the initial value, and the dashed line corresponds to the true value. The x axis for the histograms and the y axis of the chain are limits of the a priori law.

Fig. 7
Fig. 7

Joint histograms for the couple (a) ( γ 1 , w α ) and (b) ( γ 1 , w β ) . The x and y axes are the parameter values.

Fig. 8
Fig. 8

(a) Observed image and (b) restored image. Profiles correspond to the 68th line. The solid curve is the true profile, and the dashed curve correspond to (a) data and (b) estimated profiles.

Tables (4)

Tables Icon

Table 1 Error e [Eq. (46)] and Averaged Standard Deviation σ ̂ of the Posterior Image Law a

Tables Icon

Table 2 Quantitative Evaluation: True and Estimated Values of Hyperparameters and PSF Parameters

Tables Icon

Table 3 Acceptance Rate

Tables Icon

Table 4 Specific Laws Obtained As Limit of the Gamma PDF

Equations (57)

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y = H w x + ϵ ,
y ̊ = Λ H x ̊ + ϵ ̊ ,
p ( x | γ ) = ( 2 π ) N 2 det [ P ] 1 2 exp [ 1 2 x t P x ] .
p ( x | γ ) = ( 2 π ) N 2 det [ F ] det [ Λ P ] 1 2 det [ F ] exp [ 1 2 x t F Λ P F x ] ,
= ( 2 π ) N 2 det [ Λ P ] 1 2 exp [ 1 2 x ̊ Λ P x ̊ ] ,
Λ P = γ 1 Λ D Λ D = diag ( 0 , γ 1 | d ̊ 1 | 2 , , γ 1 | d ̊ N 1 | 2 ) ,
Λ P = γ 0 I + γ 1 Λ D Λ D = diag ( γ 0 , γ 0 + γ 1 | d ̊ 1 | 2 , , γ 0 + γ 1 | d ̊ N 1 | 2 )
det [ Λ P ] = n = 0 N 1 ( γ 0 + γ 1 | d ̊ n | 2 ) .
p ( x ̊ | γ 0 , γ 1 ) = p ( x ̊ 0 | γ 0 ) p ( x ̊ * | γ 0 , γ 1 ) .
Λ P = γ 0 Λ 1 Λ 1 + γ 1 Λ D Λ D . = diag ( γ 0 , γ 1 | d ̊ 1 | 2 , , γ 1 | d ̊ N 1 | 2 ) .
det [ Λ P ] = γ 0 γ 1 N 1 n = 1 N 1 | d ̊ n | 2 ,
p ( x ̊ | γ 0 , γ 1 ) = p ( x ̊ 0 | γ 0 ) p ( x ̊ * | γ 1 ) ,
p ( x | γ 0 , γ 1 ) = ( 2 π ) N 2 n = 1 N 1 | d ̊ n | γ 0 1 2 γ 1 ( N 1 ) 2 exp [ γ 0 2 x ̊ 0 2 γ 1 2 Λ D * x ̊ * 2 ] ,
p ( ϵ | γ ϵ ) = ( 2 π ) N 2 γ ϵ N 2 exp [ γ ϵ 2 ϵ 2 ] .
p ( y | x , γ ϵ , w ) = ( 2 π ) N 2 γ ϵ N 2 exp [ γ ϵ 2 y H w x 2 ] .
p ( γ i ) = 1 β i α i Γ ( α i ) γ i α i 1 exp ( γ i β i ) , γ i [ 0 , + [ .
p ( w ) = U w ¯ , δ ( w ) ,
p ( x ̊ , γ ϵ , γ 0 , γ 1 , w , y ̊ ) = p ( y ̊ | x ̊ , γ ϵ , w ) p ( x ̊ | γ 0 , γ 1 ) p ( γ ϵ ) p ( γ 0 ) p ( γ 1 ) p ( w )
p ( x ̊ , γ ϵ , γ 0 , γ 1 , w , y ̊ ) = ( 2 π ) N n = 1 N 1 | d ̊ n | β ϵ α ϵ Γ ( α ϵ ) β 0 α 0 Γ ( α 0 ) β 1 α 1 Γ ( α 1 ) γ ϵ α ϵ + N 2 1 γ 0 α 0 1 2 γ 1 α 1 + ( N 1 ) 2 1 exp [ γ ϵ β ϵ γ 0 β 0 γ 1 β 1 ] U w ¯ , δ ( w ) exp [ γ ϵ 2 y ̊ Λ H x ̊ 2 γ 0 2 x ̊ 0 2 γ 1 2 Λ D x ̊ 2 ] .
p ( x ̊ , γ ϵ , γ 0 , γ 1 , w | y ̊ ) = p ( x ̊ , γ ϵ , γ 0 , γ 1 , w , y ̊ ) p ( y ̊ ) ,
p ( y ̊ ) = p ( y ̊ , x ̊ , γ , w ) d x ̊ d γ d w .
p ( x ̊ , γ ϵ , γ 1 , w | y ̊ ) = p ( x ̊ 0 ) p ( y ̊ , x ̊ * , γ ϵ , γ 1 , w | x ̊ 0 ) p ( x ̊ 0 ) p ( y ̊ , x ̊ * , γ ϵ , γ 1 , w | x ̊ 0 ) d γ ϵ d γ 1 d w d x ̊ * d x ̊ 0
p ( x ̊ 0 ) = p ( x ̊ 0 | γ 0 ) p ( γ 0 ) d γ 0 = ( 1 + β 0 x ̊ 0 2 2 ) α 0 1 2 .
( 1 + β 0 x ̊ 0 2 2 ) α 0 1 2 p ( y ̊ , x ̊ * , γ ϵ , γ 1 , w | x ̊ 0 ) p ( y ̊ , x ̊ * , γ ϵ , γ 1 , w | x ̊ 0 ) .
R p ( y ̊ | x ̊ , γ ϵ , w ) p ( x ̊ * | γ 1 ) p ( γ 1 , γ ϵ , w ) d x ̊ 0 R p ( y ̊ 0 | x ̊ 0 , γ ϵ , w ) d x ̊ 0
R exp [ γ ϵ 2 ( y ̊ 0 h ̊ 0 x ̊ 0 ) 2 ] d x ̊ 0
p ( x ̊ , γ ϵ , γ 1 , w | y ̊ ) = p ( x ̊ , γ ϵ , γ 1 , w , y ̊ ) p ( y ̊ ) γ ϵ α ϵ + N 2 1 γ 1 α 1 + ( N 1 ) 2 1 U w ¯ , δ ( w ) exp [ γ ϵ 2 y ̊ Λ H x ̊ 2 γ 1 2 Λ D * x ̊ * 2 ] exp [ γ ϵ β ϵ γ 1 β 1 ] .
x ̊ ( k + 1 ) p ( x ̊ | y ̊ , γ ϵ ( k ) , γ 0 ( k ) , γ 1 ( k ) , w ( k ) )
N ( μ ( k + 1 ) , Σ ( k + 1 ) ) .
Σ ( k + 1 ) = ( γ ϵ ( k ) | Λ H ( k ) | 2 + γ 0 ( k ) | Λ 1 | 2 + γ 1 ( k ) | Λ D | 2 ) 1 ,
μ ( k + 1 ) = γ ϵ ( k ) Σ ( k + 1 ) Λ H ( k ) y ̊ ,
γ i ( k + 1 ) p ( γ i | y ̊ , x ̊ ( k + 1 ) , w ( k ) )
G ( γ i | α i ( k + 1 ) , β i ( k + 1 ) ) .
α ϵ ( k + 1 ) = α ϵ + N 2 and β ϵ ( k + 1 ) = ( β ϵ 1 + 1 2 y ̊ Λ H ( k ) x ̊ ( k + 1 ) 2 ) 1 ,
α 0 ( k + 1 ) = α 0 + 1 2 and β 0 ( k + 1 ) = ( β 0 1 + 1 2 ( x ̊ 0 ( k + 1 ) ) 2 ) 1 ,
α 1 ( k + 1 ) = α 1 + ( N 1 ) 2 and β 1 ( k + 1 ) = ( β 1 1 + 1 2 Λ D x ̊ ( k + 1 ) 2 ) 1 .
α ϵ ( k + 1 ) = N 2 and β ϵ ( k + 1 ) = 2 y ̊ Λ H ( k ) x ̊ ( k + 1 ) 2 ,
α 0 ( k + 1 ) = 1 2 and β 0 ( k + 1 ) = 2 ( x ̊ 0 ( k + 1 ) ) 2 ,
α 1 ( k + 1 ) = ( N 1 ) 2 and β 1 ( k + 1 ) = 2 Λ D x ̊ ( k + 1 ) 2 .
w ( k + 1 ) p ( w | y ̊ , x ̊ ( k + 1 ) , γ ϵ ( k + 1 ) )
exp [ γ ϵ ( k + 1 ) 2 y ̊ Λ H , w x ̊ ( k + 1 ) 2 ] ,
w p p ( w ) = U [ a b ] ( w ) .
J ( w ( k ) , w p ) = γ ϵ ( k + 1 ) 2 ( y ̊ Λ H , w ( k ) x ̊ ( k + 1 ) 2 y ̊ Λ H , w p x ̊ ( k + 1 ) 2 ) .
w ( k + 1 ) = { w p if log t < J w ( k ) otherwise } .
x ̂ = F E [ x ̊ ] F [ 1 K k = 0 K 1 x ̊ ( k ) ] .
h ̊ ( ν α , ν β ) = exp [ 2 π 2 ( ν α 2 ( w α cos 2 φ + w β sin 2 φ ) + ν β 2 ( w α sin 2 φ + w β cos 2 φ ) + 2 ν α ν β sin φ cos φ ( w α w β ) ) ] ,
e = x x * x *
μ ̊ = E [ x ̊ ] = F E [ x ̊ ] = F μ .
Σ ̊ = E [ ( x ̊ μ ̊ ) ( x ̊ μ ̊ ) ] = F Σ F .
Σ ̊ = F Σ F = Λ Σ ,
G ( γ | α , β ) = 1 β α Γ ( α ) γ α 1 exp ( γ β ) .
p ( x | γ ) = ( 2 π ) N 2 γ N 2 det [ Γ ] 1 2 exp [ γ x t Γ x 2 ] .
p ( x ) = R + p ( x | γ ) p ( γ ) d γ = β N 2 det [ Γ ] 1 2 Γ ( α + N 2 ) ( 2 π ) N 2 Γ ( α ) ( 1 + β x t Γ x 2 ) α N 2 ,
p ( γ | x ) = ( 2 π ) N 2 det [ Γ ] 1 2 β α Γ ( α ) γ α + N 2 1 exp [ γ x t Γ x 2 + 1 β ] .
ρ = min { f ( w p ) f ( w ( t ) ) q ( w ( t ) | w p ) q ( w p | w ( t ) ) , 1 } .
w ( t + 1 ) = { w p with ρ probability w ( t ) with 1 ρ probability } .
ρ = min { f ( w p ) f ( w ( t ) ) , 1 } .

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