Abstract

The recently proposed restoration-segmentation algorithms dedicated to polarization encoded images suffer two important limitations: the number of classes into which the image is segmented is not obtained automatically, and more importantly the quality of the segmentation is affected by the nonuniformity of the illumination of the scene. We propose here a new method addressing these issues. It is based on a global estimation-segmentation approach, explicitly modeling the nonuniform illumination. The physical admissibility of the retrieved Mueller matrices is ensured. Results stemming from synthetic and real data are provided and support the proposed approach.

© 2015 Optical Society of America

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References

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    [Crossref]
  2. J. Chung, W. Jung, M. J. Hammer-Wilson, P. Wilder-Smith, and Z. Chen, “Use of polar decomposition for the diagnosis of oral precancer,” Appl. Opt. 46, 3038–3045 (2007).
    [Crossref] [PubMed]
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  4. M. F. G. Wood, N. Ghosh, M. A. Wallenburg, S. H. Li, R. D. Weisel, B. C. Wilson, R. K. Li, and I. A. Vitkin, “Polarization birefringence Measurements for characterizing the myocardium, including healthy, infracted, and stem cell treated regenerating cardiac tissues,” J. Biomed Opt. 15, 047009 (2010).
    [Crossref]
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    [Crossref] [PubMed]
  8. S. N. Savenkov, “Optimization and structuring of the instrument matrix for polarimetric measurements,” Opt. Eng. 41, 965–972 (2002).
    [Crossref]
  9. J. Zallat, S. Anouz, and M. P. Stoll, “Optimal configurations for imaging polarimeters: impact of image noise and systematic errors,” J. Opt. A-Pure Appl. Op. 8, 807–814 (2006).
    [Crossref]
  10. S. Faisan, C. Heinrich, G. Sfikas, and J. Zallat, “Estimation of Mueller matrices using non-local means filtering,” Opt. Express 21, 4424–4438 (2013).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
  13. A. Cedilnik, K. Košmelj, and A. Blejec, “The distribution of the ratio of jointly normal variables,” Metodološki Zvezki – Advances in Methodology and Statistics 1, 99–108 (2004).
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    [Crossref] [PubMed]
  15. G. McLachlan and T. Krischnan, The EM Algorithm and Extensions, 2nd edition (Wiley, 2008).
    [Crossref]
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  18. W. H. Greene, Econometric Analysis, 7th edition (Prentice Hall, 2012), Chap. 9.
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    [Crossref]
  20. C. M. Jarque and A. K. Bera, “A test for normality of observations and regression residuals,” Int. Stat. Rev 55, 163–172 (1987).
    [Crossref]
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    [Crossref]
  22. D. Ververidis and C. Kotropoulos, “Gaussian mixture modeling by exploiting the Mahalanobis distance,” IEEE T. Signal Process 56, 2797–2811 (2008).
    [Crossref]
  23. Y. Boykov and V. Kolmogorov, “An experimental comparison of min-cut/max-flow algorithms for energy minimization in vision,” IEEE T. Pattern Anal. 26, 1124–1137 (2004).
    [Crossref]
  24. Y. Boykov, O. Veksler, and R. Zabih, “Fast approximate energy minimization via graph cuts,” IEEE T. Pattern Anal. 23, 1222–1239 (2001).
    [Crossref]
  25. V. Kolmogorov and R. Zabih, “What energy functions can be minimized via graph cuts?,,” IEEE T. Pattern Anal. 26, 147–159 (2004).
    [Crossref]

2013 (1)

2011 (1)

2010 (1)

M. F. G. Wood, N. Ghosh, M. A. Wallenburg, S. H. Li, R. D. Weisel, B. C. Wilson, R. K. Li, and I. A. Vitkin, “Polarization birefringence Measurements for characterizing the myocardium, including healthy, infracted, and stem cell treated regenerating cardiac tissues,” J. Biomed Opt. 15, 047009 (2010).
[Crossref]

2009 (1)

2008 (2)

J. Zallat, C. Heinrich, and M. Petremand, “A Bayesian approach for polarimetric data reduction: the Mueller imaging case,” Opt. Express 16, 7119–7133 (2008).
[Crossref] [PubMed]

D. Ververidis and C. Kotropoulos, “Gaussian mixture modeling by exploiting the Mahalanobis distance,” IEEE T. Signal Process 56, 2797–2811 (2008).
[Crossref]

2007 (2)

2006 (1)

J. Zallat, S. Anouz, and M. P. Stoll, “Optimal configurations for imaging polarimeters: impact of image noise and systematic errors,” J. Opt. A-Pure Appl. Op. 8, 807–814 (2006).
[Crossref]

2004 (3)

Y. Boykov and V. Kolmogorov, “An experimental comparison of min-cut/max-flow algorithms for energy minimization in vision,” IEEE T. Pattern Anal. 26, 1124–1137 (2004).
[Crossref]

V. Kolmogorov and R. Zabih, “What energy functions can be minimized via graph cuts?,,” IEEE T. Pattern Anal. 26, 147–159 (2004).
[Crossref]

A. Cedilnik, K. Košmelj, and A. Blejec, “The distribution of the ratio of jointly normal variables,” Metodološki Zvezki – Advances in Methodology and Statistics 1, 99–108 (2004).

2002 (3)

2001 (1)

Y. Boykov, O. Veksler, and R. Zabih, “Fast approximate energy minimization via graph cuts,” IEEE T. Pattern Anal. 23, 1222–1239 (2001).
[Crossref]

1995 (1)

S. Cloude and E. Pottier, “Concept of polarization entropy in optical scattering: polarization analysis and measurement,” Opt. Eng. 34, 1599–1610 (1995).
[Crossref]

1987 (1)

C. M. Jarque and A. K. Bera, “A test for normality of observations and regression residuals,” Int. Stat. Rev 55, 163–172 (1987).
[Crossref]

1985 (2)

J. A. Hartigan and P. M. Hartigan, “The dip test of unimodality,” Ann. Stat. 13, 70–84 (1985).
[Crossref]

G. Celeux and J. Diebolt, “The SEM algorithm: a probabilistic teacher algorithm derived from the EM algorithm for the mixture problem,” Comp. Statis. Quaterly 2, 73–82 (1985).

Anouz, S.

J. Zallat, S. Anouz, and M. P. Stoll, “Optimal configurations for imaging polarimeters: impact of image noise and systematic errors,” J. Opt. A-Pure Appl. Op. 8, 807–814 (2006).
[Crossref]

Antonelli, M. R.

Benali, A.

Bera, A. K.

C. M. Jarque and A. K. Bera, “A test for normality of observations and regression residuals,” Int. Stat. Rev 55, 163–172 (1987).
[Crossref]

Bilmes, J.

J. Bilmes, “A gentle tutorial of the EM algorithm and its application to parameter estimation for gaussian mixture and hidden markov models,” Tech. rep., International Computer Science Institute (1998).

Blejec, A.

A. Cedilnik, K. Košmelj, and A. Blejec, “The distribution of the ratio of jointly normal variables,” Metodološki Zvezki – Advances in Methodology and Statistics 1, 99–108 (2004).

Boykov, Y.

Y. Boykov and V. Kolmogorov, “An experimental comparison of min-cut/max-flow algorithms for energy minimization in vision,” IEEE T. Pattern Anal. 26, 1124–1137 (2004).
[Crossref]

Y. Boykov, O. Veksler, and R. Zabih, “Fast approximate energy minimization via graph cuts,” IEEE T. Pattern Anal. 23, 1222–1239 (2001).
[Crossref]

Cedilnik, A.

A. Cedilnik, K. Košmelj, and A. Blejec, “The distribution of the ratio of jointly normal variables,” Metodološki Zvezki – Advances in Methodology and Statistics 1, 99–108 (2004).

Celeux, G.

G. Celeux and J. Diebolt, “The SEM algorithm: a probabilistic teacher algorithm derived from the EM algorithm for the mixture problem,” Comp. Statis. Quaterly 2, 73–82 (1985).

Chen, Z.

Chung, J.

Cloude, S.

S. Cloude and E. Pottier, “Concept of polarization entropy in optical scattering: polarization analysis and measurement,” Opt. Eng. 34, 1599–1610 (1995).
[Crossref]

De Martino, A.

Diebolt, J.

G. Celeux and J. Diebolt, “The SEM algorithm: a probabilistic teacher algorithm derived from the EM algorithm for the mixture problem,” Comp. Statis. Quaterly 2, 73–82 (1985).

Faisan, S.

Gayet, B.

Ghosh, N.

M. F. G. Wood, N. Ghosh, M. A. Wallenburg, S. H. Li, R. D. Weisel, B. C. Wilson, R. K. Li, and I. A. Vitkin, “Polarization birefringence Measurements for characterizing the myocardium, including healthy, infracted, and stem cell treated regenerating cardiac tissues,” J. Biomed Opt. 15, 047009 (2010).
[Crossref]

M. F. G. Wood, N. Ghosh, X. Guo, and I. A. Vitkin, “Towards noninvasive glucose sensing using polarization analysis of multiply scattered light,” Chap. 17, Handbook of Optical Sensing of Glucose in Biological Fluids and Tissues, V. V. Tuchin, ed., Series in Medical Physics and Biomedical Engineering, Vol. 12 (Taylor and Francis Publishing, 2008).

Greene, W. H.

W. H. Greene, Econometric Analysis, 7th edition (Prentice Hall, 2012), Chap. 9.

Guo, X.

M. F. G. Wood, N. Ghosh, X. Guo, and I. A. Vitkin, “Towards noninvasive glucose sensing using polarization analysis of multiply scattered light,” Chap. 17, Handbook of Optical Sensing of Glucose in Biological Fluids and Tissues, V. V. Tuchin, ed., Series in Medical Physics and Biomedical Engineering, Vol. 12 (Taylor and Francis Publishing, 2008).

Hammer-Wilson, M. J.

Hartigan, J. A.

J. A. Hartigan and P. M. Hartigan, “The dip test of unimodality,” Ann. Stat. 13, 70–84 (1985).
[Crossref]

Hartigan, P. M.

J. A. Hartigan and P. M. Hartigan, “The dip test of unimodality,” Ann. Stat. 13, 70–84 (1985).
[Crossref]

Hasinoff, S. W.

S. W. Hasinoff, “ Photon, Poisson Noise,” in Computer Vision, Katsushi Ikeuchi, ed. (SpringerUS, 2014)
[Crossref]

Heinrich, C.

Ikeuchi, K.

Jarque, C. M.

C. M. Jarque and A. K. Bera, “A test for normality of observations and regression residuals,” Int. Stat. Rev 55, 163–172 (1987).
[Crossref]

Jung, W.

Kolmogorov, V.

V. Kolmogorov and R. Zabih, “What energy functions can be minimized via graph cuts?,,” IEEE T. Pattern Anal. 26, 147–159 (2004).
[Crossref]

Y. Boykov and V. Kolmogorov, “An experimental comparison of min-cut/max-flow algorithms for energy minimization in vision,” IEEE T. Pattern Anal. 26, 1124–1137 (2004).
[Crossref]

Košmelj, K.

A. Cedilnik, K. Košmelj, and A. Blejec, “The distribution of the ratio of jointly normal variables,” Metodološki Zvezki – Advances in Methodology and Statistics 1, 99–108 (2004).

Kotropoulos, C.

D. Ververidis and C. Kotropoulos, “Gaussian mixture modeling by exploiting the Mahalanobis distance,” IEEE T. Signal Process 56, 2797–2811 (2008).
[Crossref]

Krischnan, T.

G. McLachlan and T. Krischnan, The EM Algorithm and Extensions, 2nd edition (Wiley, 2008).
[Crossref]

Li, R. K.

M. F. G. Wood, N. Ghosh, M. A. Wallenburg, S. H. Li, R. D. Weisel, B. C. Wilson, R. K. Li, and I. A. Vitkin, “Polarization birefringence Measurements for characterizing the myocardium, including healthy, infracted, and stem cell treated regenerating cardiac tissues,” J. Biomed Opt. 15, 047009 (2010).
[Crossref]

Li, S. H.

M. F. G. Wood, N. Ghosh, M. A. Wallenburg, S. H. Li, R. D. Weisel, B. C. Wilson, R. K. Li, and I. A. Vitkin, “Polarization birefringence Measurements for characterizing the myocardium, including healthy, infracted, and stem cell treated regenerating cardiac tissues,” J. Biomed Opt. 15, 047009 (2010).
[Crossref]

McLachlan, G.

G. McLachlan and T. Krischnan, The EM Algorithm and Extensions, 2nd edition (Wiley, 2008).
[Crossref]

Miyasaki, D.

Novikova, T.

Petremand, M.

Pierangelo, A.

Pottier, E.

S. Cloude and E. Pottier, “Concept of polarization entropy in optical scattering: polarization analysis and measurement,” Opt. Eng. 34, 1599–1610 (1995).
[Crossref]

Pradhan, A.

Saito, M.

Sato, Y.

Savenkov, S. N.

S. N. Savenkov, “Optimization and structuring of the instrument matrix for polarimetric measurements,” Opt. Eng. 41, 965–972 (2002).
[Crossref]

Sfikas, G.

Shukla, P.

Stoll, M. P.

J. Zallat, S. Anouz, and M. P. Stoll, “Optimal configurations for imaging polarimeters: impact of image noise and systematic errors,” J. Opt. A-Pure Appl. Op. 8, 807–814 (2006).
[Crossref]

Tyo, J. S.

Validire, P.

Veksler, O.

Y. Boykov, O. Veksler, and R. Zabih, “Fast approximate energy minimization via graph cuts,” IEEE T. Pattern Anal. 23, 1222–1239 (2001).
[Crossref]

Ververidis, D.

D. Ververidis and C. Kotropoulos, “Gaussian mixture modeling by exploiting the Mahalanobis distance,” IEEE T. Signal Process 56, 2797–2811 (2008).
[Crossref]

Vitkin, I. A.

M. F. G. Wood, N. Ghosh, M. A. Wallenburg, S. H. Li, R. D. Weisel, B. C. Wilson, R. K. Li, and I. A. Vitkin, “Polarization birefringence Measurements for characterizing the myocardium, including healthy, infracted, and stem cell treated regenerating cardiac tissues,” J. Biomed Opt. 15, 047009 (2010).
[Crossref]

M. F. G. Wood, N. Ghosh, X. Guo, and I. A. Vitkin, “Towards noninvasive glucose sensing using polarization analysis of multiply scattered light,” Chap. 17, Handbook of Optical Sensing of Glucose in Biological Fluids and Tissues, V. V. Tuchin, ed., Series in Medical Physics and Biomedical Engineering, Vol. 12 (Taylor and Francis Publishing, 2008).

Wallenburg, M. A.

M. F. G. Wood, N. Ghosh, M. A. Wallenburg, S. H. Li, R. D. Weisel, B. C. Wilson, R. K. Li, and I. A. Vitkin, “Polarization birefringence Measurements for characterizing the myocardium, including healthy, infracted, and stem cell treated regenerating cardiac tissues,” J. Biomed Opt. 15, 047009 (2010).
[Crossref]

Weisel, R. D.

M. F. G. Wood, N. Ghosh, M. A. Wallenburg, S. H. Li, R. D. Weisel, B. C. Wilson, R. K. Li, and I. A. Vitkin, “Polarization birefringence Measurements for characterizing the myocardium, including healthy, infracted, and stem cell treated regenerating cardiac tissues,” J. Biomed Opt. 15, 047009 (2010).
[Crossref]

Wilder-Smith, P.

Wilson, B. C.

M. F. G. Wood, N. Ghosh, M. A. Wallenburg, S. H. Li, R. D. Weisel, B. C. Wilson, R. K. Li, and I. A. Vitkin, “Polarization birefringence Measurements for characterizing the myocardium, including healthy, infracted, and stem cell treated regenerating cardiac tissues,” J. Biomed Opt. 15, 047009 (2010).
[Crossref]

Wood, M. F. G.

M. F. G. Wood, N. Ghosh, M. A. Wallenburg, S. H. Li, R. D. Weisel, B. C. Wilson, R. K. Li, and I. A. Vitkin, “Polarization birefringence Measurements for characterizing the myocardium, including healthy, infracted, and stem cell treated regenerating cardiac tissues,” J. Biomed Opt. 15, 047009 (2010).
[Crossref]

M. F. G. Wood, N. Ghosh, X. Guo, and I. A. Vitkin, “Towards noninvasive glucose sensing using polarization analysis of multiply scattered light,” Chap. 17, Handbook of Optical Sensing of Glucose in Biological Fluids and Tissues, V. V. Tuchin, ed., Series in Medical Physics and Biomedical Engineering, Vol. 12 (Taylor and Francis Publishing, 2008).

Zabih, R.

V. Kolmogorov and R. Zabih, “What energy functions can be minimized via graph cuts?,,” IEEE T. Pattern Anal. 26, 147–159 (2004).
[Crossref]

Y. Boykov, O. Veksler, and R. Zabih, “Fast approximate energy minimization via graph cuts,” IEEE T. Pattern Anal. 23, 1222–1239 (2001).
[Crossref]

Zallat, J.

Ann. Stat. (1)

J. A. Hartigan and P. M. Hartigan, “The dip test of unimodality,” Ann. Stat. 13, 70–84 (1985).
[Crossref]

Appl. Opt. (2)

Comp. Statis. Quaterly (1)

G. Celeux and J. Diebolt, “The SEM algorithm: a probabilistic teacher algorithm derived from the EM algorithm for the mixture problem,” Comp. Statis. Quaterly 2, 73–82 (1985).

IEEE T. Pattern Anal. (3)

Y. Boykov and V. Kolmogorov, “An experimental comparison of min-cut/max-flow algorithms for energy minimization in vision,” IEEE T. Pattern Anal. 26, 1124–1137 (2004).
[Crossref]

Y. Boykov, O. Veksler, and R. Zabih, “Fast approximate energy minimization via graph cuts,” IEEE T. Pattern Anal. 23, 1222–1239 (2001).
[Crossref]

V. Kolmogorov and R. Zabih, “What energy functions can be minimized via graph cuts?,,” IEEE T. Pattern Anal. 26, 147–159 (2004).
[Crossref]

IEEE T. Signal Process (1)

D. Ververidis and C. Kotropoulos, “Gaussian mixture modeling by exploiting the Mahalanobis distance,” IEEE T. Signal Process 56, 2797–2811 (2008).
[Crossref]

Int. Stat. Rev (1)

C. M. Jarque and A. K. Bera, “A test for normality of observations and regression residuals,” Int. Stat. Rev 55, 163–172 (1987).
[Crossref]

J. Biomed Opt. (1)

M. F. G. Wood, N. Ghosh, M. A. Wallenburg, S. H. Li, R. D. Weisel, B. C. Wilson, R. K. Li, and I. A. Vitkin, “Polarization birefringence Measurements for characterizing the myocardium, including healthy, infracted, and stem cell treated regenerating cardiac tissues,” J. Biomed Opt. 15, 047009 (2010).
[Crossref]

J. Opt. A-Pure Appl. Op. (1)

J. Zallat, S. Anouz, and M. P. Stoll, “Optimal configurations for imaging polarimeters: impact of image noise and systematic errors,” J. Opt. A-Pure Appl. Op. 8, 807–814 (2006).
[Crossref]

J. Opt. Soc. Am. A (1)

Metodološki Zvezki – Advances in Methodology and Statistics (1)

A. Cedilnik, K. Košmelj, and A. Blejec, “The distribution of the ratio of jointly normal variables,” Metodološki Zvezki – Advances in Methodology and Statistics 1, 99–108 (2004).

Opt. Eng. (2)

S. N. Savenkov, “Optimization and structuring of the instrument matrix for polarimetric measurements,” Opt. Eng. 41, 965–972 (2002).
[Crossref]

S. Cloude and E. Pottier, “Concept of polarization entropy in optical scattering: polarization analysis and measurement,” Opt. Eng. 34, 1599–1610 (1995).
[Crossref]

Opt. Express (5)

Other (5)

M. F. G. Wood, N. Ghosh, X. Guo, and I. A. Vitkin, “Towards noninvasive glucose sensing using polarization analysis of multiply scattered light,” Chap. 17, Handbook of Optical Sensing of Glucose in Biological Fluids and Tissues, V. V. Tuchin, ed., Series in Medical Physics and Biomedical Engineering, Vol. 12 (Taylor and Francis Publishing, 2008).

S. W. Hasinoff, “ Photon, Poisson Noise,” in Computer Vision, Katsushi Ikeuchi, ed. (SpringerUS, 2014)
[Crossref]

G. McLachlan and T. Krischnan, The EM Algorithm and Extensions, 2nd edition (Wiley, 2008).
[Crossref]

J. Bilmes, “A gentle tutorial of the EM algorithm and its application to parameter estimation for gaussian mixture and hidden markov models,” Tech. rep., International Computer Science Institute (1998).

W. H. Greene, Econometric Analysis, 7th edition (Prentice Hall, 2012), Chap. 9.

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Figures (5)

Fig. 1
Fig. 1 Steps related to the estimation of the parameters Θ and of the number K of classes.
Fig. 2
Fig. 2 (a) Simulated segmentation map, (b) simulated lighting map, and (c) example of a mean transmittance image obtained for a 10 dB SNR. Representative segmentation maps obtained (SNR=10 dB) with the proposed approach (d), and with the approach of [14] (without normalization (e), by using the pseudo-inverse for normalization (thresholding (f) and polynomial modeling (g)), and by using the ground truth for normalization (h).
Fig. 3
Fig. 3 Exemple of synthetic intensity images obtained with a SNR of 10 dB.
Fig. 4
Fig. 4 Sketch of the scene composition (a), automatic segmentation with the proposed approach using the Jarque–Bera test (7 classes) (b), and using the dip test (c).
Fig. 5
Fig. 5 The 16 intensity images of the object acquired by a rotating quarter–wave plate Mueller imaging polarimeter in transmission configuration.

Tables (4)

Tables Icon

Table 1 Mueller matrices used for the creation of the synthetic data.

Tables Icon

Table 2 Number of occurrences of the estimated number K of classes for 100 experiments, for different values of α and of the SNR, with the Jarque-Bera test.

Tables Icon

Table 3 Mean values of the relative estimation error RE (in percent) for the four Mueller matrices and the corresponding covariance matrices. Averaging has been done over 100 realizations. Numbers between parentheses represent standard-deviation of related quantities.

Tables Icon

Table 4 Mueller matrices associated to dichroic patches and to their related linear polarizer.

Equations (45)

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

I ( s ) = P M ( s ) _ ( + noise ) ,
I ( s ) = κ ( s ) P M k _ + ε ( s ) = κ ( s ) Ψ k + ε ( s ) ,
P ( I ( s ) , y ( s ) = k , κ ( s ) | Θ ) = P ( I ( s ) | y ( s ) = k , κ ( s ) , Θ ) P ( y ( s ) = k | Θ ) P ( κ ( s ) | Θ ) .
I ( s ) | y ( s ) = k , κ ( s ) , Θ ~ 𝒩 ( κ ( s ) Ψ k , Σ k ) .
P ( I ( s ) | Θ ) = k = 1 K κ ( s ) P ( I ( s ) , y ( s ) = k , κ ( s ) | Θ ) d κ ( s ) ,
P ( I ( s ) | Θ ) = k = 1 K κ ( s ) π k P ( κ ( s ) | Θ ) P ( I ( s ) | y ( s ) = k , κ ( s ) , Θ ) d κ ( s ) .
L ( Θ ) = P ( I | Θ ) = s P ( I ( s ) | Θ ) .
P ( I , y , κ | Θ ) = P ( I , y , κ | Θ ) P ( y , κ | Θ ) = [ s P ( I ( s ) | y ( s ) , κ ( s ) | Θ ) ] P ( y | Θ ) P ( κ | Θ ) = [ s P ( I ( s ) | y ( s ) , κ ( s ) , Θ ) ] [ s P ( y ( s ) | Θ ) ] [ s P ( κ ( s ) | Θ ) ] = s P ( I ( s ) , y ( s ) , κ ( s ) | Θ ) ,
Q ( Θ , Θ ) = s k = 1 K w k ( s ) κ ( s ) = 0 + log [ P ( I ( s ) , y ( s ) = k , κ ( s ) | Θ k ) ] × P ( κ ( s ) | y ( s ) = k , I ( s ) , Θ ) d κ ( s ) ,
κ k ( s ) | y ( s ) = k , I ( s ) , Θ ~ 𝒩 + ( α k β k , α k ) ,
Θ ^ = arg max Θ s k = 1 K w k ( s ) log [ P ( I ( s ) , y ( s ) = k , k κ ( s ) | Θ k ) ] = arg max Θ s k = 1 K w k ( s ) log [ π k P ( I ( s ) | Θ k , y ( s ) = k , k κ ( s ) ) ]
π ^ k = 1 # pixels s w k ( s ) ,
arg min { Ψ , Σ } s w k ( s ) [ log ( | Σ | ) + ( I ( s ) κ k ( s ) Ψ ) T Σ 1 ( I ( s ) κ k ( s ) Ψ ) ] .
I ¯ k = 1 s w k ( s ) κ k 2 ( s ) s w k ( s ) κ k ( s ) I ( s ) .
M ^ k = arg min M Σ k 1 / 2 ( I ¯ k P M ) 2 ,
Σ ^ k = ( 1 s w k ( s ) s w k ( s ) ( I ( s ) κ k ( s ) Ψ ^ k ) ( I ( s ) κ k ( s ) Ψ ^ k ) T ) I N × N ,
r ^ k ( s ) = I ( s ) κ ^ k ( s ) Ψ ^ k N
p min = min { p k , n } k = 1 K , n = 1 N .
p min 1 ( 1 α ) 1 N K .
P ( I 0 | β ) = 1 Z ( β ) exp ( β U ( I 0 ) ) ,
U ( I 0 ) = [ t , s ] 𝕀 ( I 0 ( t ) = I 0 ( s ) )
{ I ^ 0 , β ^ } = arg max { I 0 , β } log P ( I 0 , β | I , Θ ) = arg max { I 0 , β } log P ( I 0 | β ) + s log P ( I ( s ) | I 0 ( s ) , Θ ) ,
{ P ( I ( s ) , κ ( s ) | I 0 ( s ) , Θ ) = P ( I ( s ) | I 0 ( s ) , κ ( s ) , Θ ) P ( κ ( s ) ) , I ( s ) | y ( s ) = k , κ ( s ) , Θ ~ 𝒩 ( κ ( s ) Ψ k , Σ k ) , P ( I ( s ) | I 0 ( s ) , Θ ) = P ( I ( s ) , κ ( s ) | I 0 ( s ) , Θ ) d κ ( s ) ,
RE Mk = M ^ k M k M k , RE Σ k = Σ ^ k Σ k Σ k ,
I norm ( s ) = κ ( s ) m 00 ( s ) P M k _ + ε ( s ) m 00 ( s ) , P M k _ + ε ( s ) m 00 ( s ) .
I norm ( s ) = P M k _ + ε ( s ) κ ( s ) ,
Ψ 1 = s I ( s ) s κ ^ ( s ) and M 1 = ( P t P ) 1 P t Ψ 1 .
π K + 1 = π k ( p ) = s C K + 1 w k ( p ) ( s ) s C K + 1 C k w k ( p ) ( s ) , and π k = π k ( p ) s C k w k ( p ) ( s ) s C K + 1 C k w k ( p ) ( s )
log P ( I , y , κ | Θ ) = s log [ P ( I ( s ) , y ( s ) , κ ( s ) | Θ ) ] .
Q ( Θ , Θ ) = y κ log [ P ( I , y , κ | Θ ) ] P ( y , κ | I , Θ ) d κ .
Q ( Θ , Θ ) = s k = 1 K κ ( s ) = 0 + log [ P ( I ( s ) , y ( s ) = k , κ ( s ) | Θ k ) ] P ( y ( s ) = k , κ ( s ) | I ( s ) , Θ ) d κ ( s ) .
P ( y ( s ) = k , κ ( s ) | I ( s ) , Θ ) = P ( κ ( s ) | y ( s ) = k , I ( s ) , Θ ) P ( y ( s ) = k | I ( s ) , Θ ) ,
w k ( s ) = P ( y ( s ) = k | I ( s ) , Θ ) P ( y ( s ) = k , I ( s ) | Θ ) P ( I ( s ) | y ( s ) = k , Θ ) P ( y ( s ) = k | Θ ) [ P ( I ( s ) , κ ( s ) | y ( s ) = k , Θ ) d κ ( s ) ] π k π k κ ( s ) = 0 + P ( I ( s ) | y ( s ) = k , κ ( s ) , Θ k ) d κ ( s ) π k α k | Σ k | exp ( α k β k 2 γ k 2 ) ( 1 + erf ( β k α k 2 ) ) ,
{ α k = ( Ψ k T Σ k 1 Ψ k ) 1 β k = Ψ k T Σ k 1 I ( s ) γ k = I ( s ) Σ k 1 I ( s ) .
Q ( Θ , Θ ) = s k = 1 K w k ( s ) κ ( s ) = 0 + log [ P ( I ( s ) , y ( s ) = k , κ ( s ) | Θ k ) ] × P ( κ ( s ) | y ( s ) = k , I ( s ) , Θ ) d κ ( s ) .
I ¯ k = 1 s w k ( s ) κ k 2 ( s ) s w k ( s ) κ k ( s ) I ( s ) .
{ M ^ k , Σ ^ k } = arg min Θ = { M , Σ } ( s w k ( s ) ) log ( | Σ | ) + ( s w k ( s ) κ k 2 ( s ) ) Σ 1 / 2 ( I ¯ k P M ) 2 .
M ^ k = arg min M Σ k 1 / 2 ( I ¯ k PM ) 2 ,
Σ ^ k = ( 1 s w k ( s ) s w k ( s ) ( I ( s ) κ k ( s ) Ψ ^ k ) ( I ( s ) κ k ( s ) Ψ ^ k ) T ) I N × N ,
c k ( Ψ , Σ ) = s w k ( s ) ( I ( s ) κ k ( s ) Ψ ) T Σ 1 ( I ( s ) κ k ( s ) Ψ ) .
c k ( Ψ , Σ ) = s w k ( s ) κ k 2 ( s ) ( I ( 1 ) ( s ) Ψ ( 1 ) ) T ( I ( 1 ) ( s ) Ψ ( 1 ) ) ) ,
D ( s ) = w k ( s ) κ k 2 ( s ) s w k ( s ) κ k 2 ( s ) I N × N ,
c k ( Ψ , Σ ) = ( s w k ( s ) κ k 2 ( s ) ) s ( I ( 1 ) ( s ) Ψ ( 1 ) ) T D ( s ) ( I ( 1 ) ( s ) Ψ ( 1 ) ) .
s ( I ( 1 ) ( s ) P ( 1 ) M ) T D ( s ) ( I ( 1 ) ( s ) P ( 1 ) M ) = I ¯ P ( 1 ) M 2 ( + constant term ) ,
c k ( Ψ , Σ ) = ( s w k ( s ) κ k 2 ( s ) ) Σ 1 / 2 ( I ¯ k P M ) 2 ( + constant term )

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