Abstract

A Bayesian approach for joint restoration and segmentation of polarization encoded images is presented with emphasis on both physical admissibility and smoothness of the solution. Two distinct models for the sought polarized radiances are used: (i) the polarized light at each site of the image is described by its Stokes vector, which directly follows a mixture of truncated Gaussians, explicitly assigning zero probability to inadmissible configurations and (ii) polarization at each site is represented by the coherency matrix, which is parameterized by a set of variables assumed to be generated by a spatially varying mixture of Gaussians. Application on real and synthetic images using the proposed methods assesses the pertinence of the approach.

© 2011 Optical Society of America

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  1. R. Molina, J. Mateos, A. Katsaggelos, and M. Vega, “Bayesian multichannel image restoration using compound Gauss-Markov random fields,” IEEE Trans. Image Process. 12, 1642-1654(2003).
    [CrossRef]
  2. J. Chantas, N. Galatsanos, and A. Likas, “Bayesian restoration using a new hierarchical directional continuous edge image prior,” IEEE Trans. Image Process. 15, 2987-2997 (2006).
    [CrossRef] [PubMed]
  3. L. Bar, A. Brook, N. Sochen, and N. Kiryati, “Deblurring of color images corrupted by impulsive noise,” IEEE Trans. Image Process. 16, 1101-1111 (2007).
    [CrossRef] [PubMed]
  4. J. Zallat and C. Heinrich, “Polarimetric data reduction: a Bayesian approach,” Opt. Express 15, 83-96 (2007).
    [CrossRef] [PubMed]
  5. J. R. Valenzuela and J. A. Fessler, “Joint reconstruction of Stokes images from polarimetric measurements,” J. Opt. Soc. Am. A 26, 962-968 (2009).
    [CrossRef]
  6. J. R. Valenzuela, J. A. Fessler, and R. G. Paxman, “Joint estimation of Stokes images and aberrations from phase-diverse polarimetric measurements,” J. Opt. Soc. Am. A 27, 1185-1193(2010).
    [CrossRef]
  7. D. LeMaster and S. Cain, “Multichannel blind deconvolution of polarimetric imagery,” J. Opt. Soc. Am. A 25, 2170-2176(2008).
    [CrossRef]
  8. J. Zallat, C. Collet, and Y. Takakura, “Clustering of polarization-encoded images,” Appl. Opt. 43, 283-292 (2004).
    [CrossRef] [PubMed]
  9. G. Sfikas, C. Heinrich, J. Zallat, C. Nikou, and N. Galatsanos, “Joint recovery and segmentation of polarimetric images using a compound mrf and mixture modeling,” in Proceedings of the IEEE International Conference on Image Processing (ICIP 09) (IEEE, 2009), pp. 3901-3904.
    [CrossRef]
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    [CrossRef]
  13. S. Geman and D. Geman, “Stochastic relaxation, Gibbs distribution and the Bayesian restoration of images,” IEEE Trans. Pattern Anal. Machine Intell. PAMI-6, 721-741 (1984).
    [CrossRef]
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    [CrossRef] [PubMed]
  15. G. McLachlan, Finite Mixture Models (Wiley-Interscience, 2000).
    [CrossRef]
  16. C. M. Bishop, Pattern Recognition and Machine Learning. (Springer, 2006).
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    [CrossRef]
  18. J. Zallat, S. Aïnouz, and M. Stoll, “Optimal configurations for imaging polarimeters: impact of image noise and systematic errors,” J. Opt. A Pure Appl. Opt. 8, 807-814 (2006).
    [CrossRef]
  19. 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]
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    [CrossRef]

2010 (2)

J. R. Valenzuela, J. A. Fessler, and R. G. Paxman, “Joint estimation of Stokes images and aberrations from phase-diverse polarimetric measurements,” J. Opt. Soc. Am. A 27, 1185-1193(2010).
[CrossRef]

G. Sfikas, C. Nikou, N. Galatsanos, and C. Heinrich, “Spatially varying mixtures incorporating line processes for image segmentation,” J. Math Imaging Vis. 36, 91-110 (2010).
[CrossRef]

2009 (1)

2008 (2)

2007 (3)

L. Bar, A. Brook, N. Sochen, and N. Kiryati, “Deblurring of color images corrupted by impulsive noise,” IEEE Trans. Image Process. 16, 1101-1111 (2007).
[CrossRef] [PubMed]

J. Zallat and C. Heinrich, “Polarimetric data reduction: a Bayesian approach,” Opt. Express 15, 83-96 (2007).
[CrossRef] [PubMed]

C. Nikou, N. Galatsanos, and A. Likas, “A class-adaptive spatially variant mixture model for image segmentation,” IEEE Trans. Image Process. 16, 1121-1130 (2007).
[CrossRef] [PubMed]

2006 (2)

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

J. Chantas, N. Galatsanos, and A. Likas, “Bayesian restoration using a new hierarchical directional continuous edge image prior,” IEEE Trans. Image Process. 15, 2987-2997 (2006).
[CrossRef] [PubMed]

2004 (1)

2003 (1)

R. Molina, J. Mateos, A. Katsaggelos, and M. Vega, “Bayesian multichannel image restoration using compound Gauss-Markov random fields,” IEEE Trans. Image Process. 12, 1642-1654(2003).
[CrossRef]

2002 (1)

B. Aiazzi, L. Alparone, A. Barducci, S. Baronti, and I. Pippi, “Estimating noise and information of multispectral imagery,” Opt. Eng. 41, 656-668 (2002).
[CrossRef]

1998 (1)

J. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder-Mead simplex method in low dimensions,” SIAM J. Optim. 9, 112-147 (1998).
[CrossRef]

1984 (1)

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

1977 (1)

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

Aiazzi, B.

B. Aiazzi, L. Alparone, A. Barducci, S. Baronti, and I. Pippi, “Estimating noise and information of multispectral imagery,” Opt. Eng. 41, 656-668 (2002).
[CrossRef]

Aïnouz, S.

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

Alparone, L.

B. Aiazzi, L. Alparone, A. Barducci, S. Baronti, and I. Pippi, “Estimating noise and information of multispectral imagery,” Opt. Eng. 41, 656-668 (2002).
[CrossRef]

Bar, L.

L. Bar, A. Brook, N. Sochen, and N. Kiryati, “Deblurring of color images corrupted by impulsive noise,” IEEE Trans. Image Process. 16, 1101-1111 (2007).
[CrossRef] [PubMed]

Barducci, A.

B. Aiazzi, L. Alparone, A. Barducci, S. Baronti, and I. Pippi, “Estimating noise and information of multispectral imagery,” Opt. Eng. 41, 656-668 (2002).
[CrossRef]

Baronti, S.

B. Aiazzi, L. Alparone, A. Barducci, S. Baronti, and I. Pippi, “Estimating noise and information of multispectral imagery,” Opt. Eng. 41, 656-668 (2002).
[CrossRef]

Birkhoff, G.

G. Birkhoff and S. MacLane, A Survey of Modern Algebra (McMillan, 1953).

Bishop, C. M.

C. M. Bishop, Pattern Recognition and Machine Learning. (Springer, 2006).

Brook, A.

L. Bar, A. Brook, N. Sochen, and N. Kiryati, “Deblurring of color images corrupted by impulsive noise,” IEEE Trans. Image Process. 16, 1101-1111 (2007).
[CrossRef] [PubMed]

Cain, S.

Chantas, J.

J. Chantas, N. Galatsanos, and A. Likas, “Bayesian restoration using a new hierarchical directional continuous edge image prior,” IEEE Trans. Image Process. 15, 2987-2997 (2006).
[CrossRef] [PubMed]

Collet, C.

Dempster, A. P.

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

Fessler, J. A.

Galatsanos, N.

G. Sfikas, C. Nikou, N. Galatsanos, and C. Heinrich, “Spatially varying mixtures incorporating line processes for image segmentation,” J. Math Imaging Vis. 36, 91-110 (2010).
[CrossRef]

C. Nikou, N. Galatsanos, and A. Likas, “A class-adaptive spatially variant mixture model for image segmentation,” IEEE Trans. Image Process. 16, 1121-1130 (2007).
[CrossRef] [PubMed]

J. Chantas, N. Galatsanos, and A. Likas, “Bayesian restoration using a new hierarchical directional continuous edge image prior,” IEEE Trans. Image Process. 15, 2987-2997 (2006).
[CrossRef] [PubMed]

G. Sfikas, C. Heinrich, J. Zallat, C. Nikou, and N. Galatsanos, “Joint recovery and segmentation of polarimetric images using a compound mrf and mixture modeling,” in Proceedings of the IEEE International Conference on Image Processing (ICIP 09) (IEEE, 2009), pp. 3901-3904.
[CrossRef]

Geman, D.

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

Geman, S.

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

Heinrich, C.

G. Sfikas, C. Nikou, N. Galatsanos, and C. Heinrich, “Spatially varying mixtures incorporating line processes for image segmentation,” J. Math Imaging Vis. 36, 91-110 (2010).
[CrossRef]

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]

J. Zallat and C. Heinrich, “Polarimetric data reduction: a Bayesian approach,” Opt. Express 15, 83-96 (2007).
[CrossRef] [PubMed]

G. Sfikas, C. Heinrich, J. Zallat, C. Nikou, and N. Galatsanos, “Joint recovery and segmentation of polarimetric images using a compound mrf and mixture modeling,” in Proceedings of the IEEE International Conference on Image Processing (ICIP 09) (IEEE, 2009), pp. 3901-3904.
[CrossRef]

Katsaggelos, A.

R. Molina, J. Mateos, A. Katsaggelos, and M. Vega, “Bayesian multichannel image restoration using compound Gauss-Markov random fields,” IEEE Trans. Image Process. 12, 1642-1654(2003).
[CrossRef]

Kiryati, N.

L. Bar, A. Brook, N. Sochen, and N. Kiryati, “Deblurring of color images corrupted by impulsive noise,” IEEE Trans. Image Process. 16, 1101-1111 (2007).
[CrossRef] [PubMed]

Lagarias, J.

J. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder-Mead simplex method in low dimensions,” SIAM J. Optim. 9, 112-147 (1998).
[CrossRef]

Laird, N. M.

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

Lange, K.

K. Lange, Optimization (Springer, 2004).

LeMaster, D.

Likas, A.

C. Nikou, N. Galatsanos, and A. Likas, “A class-adaptive spatially variant mixture model for image segmentation,” IEEE Trans. Image Process. 16, 1121-1130 (2007).
[CrossRef] [PubMed]

J. Chantas, N. Galatsanos, and A. Likas, “Bayesian restoration using a new hierarchical directional continuous edge image prior,” IEEE Trans. Image Process. 15, 2987-2997 (2006).
[CrossRef] [PubMed]

MacLane, S.

G. Birkhoff and S. MacLane, A Survey of Modern Algebra (McMillan, 1953).

Mateos, J.

R. Molina, J. Mateos, A. Katsaggelos, and M. Vega, “Bayesian multichannel image restoration using compound Gauss-Markov random fields,” IEEE Trans. Image Process. 12, 1642-1654(2003).
[CrossRef]

McLachlan, G.

G. McLachlan, Finite Mixture Models (Wiley-Interscience, 2000).
[CrossRef]

Molina, R.

R. Molina, J. Mateos, A. Katsaggelos, and M. Vega, “Bayesian multichannel image restoration using compound Gauss-Markov random fields,” IEEE Trans. Image Process. 12, 1642-1654(2003).
[CrossRef]

Nikou, C.

G. Sfikas, C. Nikou, N. Galatsanos, and C. Heinrich, “Spatially varying mixtures incorporating line processes for image segmentation,” J. Math Imaging Vis. 36, 91-110 (2010).
[CrossRef]

C. Nikou, N. Galatsanos, and A. Likas, “A class-adaptive spatially variant mixture model for image segmentation,” IEEE Trans. Image Process. 16, 1121-1130 (2007).
[CrossRef] [PubMed]

G. Sfikas, C. Heinrich, J. Zallat, C. Nikou, and N. Galatsanos, “Joint recovery and segmentation of polarimetric images using a compound mrf and mixture modeling,” in Proceedings of the IEEE International Conference on Image Processing (ICIP 09) (IEEE, 2009), pp. 3901-3904.
[CrossRef]

Paxman, R. G.

Petremand, M.

Pippi, I.

B. Aiazzi, L. Alparone, A. Barducci, S. Baronti, and I. Pippi, “Estimating noise and information of multispectral imagery,” Opt. Eng. 41, 656-668 (2002).
[CrossRef]

Reeds, J. A.

J. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder-Mead simplex method in low dimensions,” SIAM J. Optim. 9, 112-147 (1998).
[CrossRef]

Rubin, D. B.

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

Sfikas, G.

G. Sfikas, C. Nikou, N. Galatsanos, and C. Heinrich, “Spatially varying mixtures incorporating line processes for image segmentation,” J. Math Imaging Vis. 36, 91-110 (2010).
[CrossRef]

G. Sfikas, C. Heinrich, J. Zallat, C. Nikou, and N. Galatsanos, “Joint recovery and segmentation of polarimetric images using a compound mrf and mixture modeling,” in Proceedings of the IEEE International Conference on Image Processing (ICIP 09) (IEEE, 2009), pp. 3901-3904.
[CrossRef]

Sochen, N.

L. Bar, A. Brook, N. Sochen, and N. Kiryati, “Deblurring of color images corrupted by impulsive noise,” IEEE Trans. Image Process. 16, 1101-1111 (2007).
[CrossRef] [PubMed]

Stoll, M.

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

Takakura, Y.

Valenzuela, J. R.

Vega, M.

R. Molina, J. Mateos, A. Katsaggelos, and M. Vega, “Bayesian multichannel image restoration using compound Gauss-Markov random fields,” IEEE Trans. Image Process. 12, 1642-1654(2003).
[CrossRef]

Wright, M. H.

J. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder-Mead simplex method in low dimensions,” SIAM J. Optim. 9, 112-147 (1998).
[CrossRef]

Wright, P. E.

J. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder-Mead simplex method in low dimensions,” SIAM J. Optim. 9, 112-147 (1998).
[CrossRef]

Zallat, J.

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]

J. Zallat and C. Heinrich, “Polarimetric data reduction: a Bayesian approach,” Opt. Express 15, 83-96 (2007).
[CrossRef] [PubMed]

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

J. Zallat, C. Collet, and Y. Takakura, “Clustering of polarization-encoded images,” Appl. Opt. 43, 283-292 (2004).
[CrossRef] [PubMed]

G. Sfikas, C. Heinrich, J. Zallat, C. Nikou, and N. Galatsanos, “Joint recovery and segmentation of polarimetric images using a compound mrf and mixture modeling,” in Proceedings of the IEEE International Conference on Image Processing (ICIP 09) (IEEE, 2009), pp. 3901-3904.
[CrossRef]

Appl. Opt. (1)

IEEE Trans. Image Process. (4)

R. Molina, J. Mateos, A. Katsaggelos, and M. Vega, “Bayesian multichannel image restoration using compound Gauss-Markov random fields,” IEEE Trans. Image Process. 12, 1642-1654(2003).
[CrossRef]

J. Chantas, N. Galatsanos, and A. Likas, “Bayesian restoration using a new hierarchical directional continuous edge image prior,” IEEE Trans. Image Process. 15, 2987-2997 (2006).
[CrossRef] [PubMed]

L. Bar, A. Brook, N. Sochen, and N. Kiryati, “Deblurring of color images corrupted by impulsive noise,” IEEE Trans. Image Process. 16, 1101-1111 (2007).
[CrossRef] [PubMed]

C. Nikou, N. Galatsanos, and A. Likas, “A class-adaptive spatially variant mixture model for image segmentation,” IEEE Trans. Image Process. 16, 1121-1130 (2007).
[CrossRef] [PubMed]

IEEE Trans. Pattern Anal. Machine Intell. (1)

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

J. Math Imaging Vis. (1)

G. Sfikas, C. Nikou, N. Galatsanos, and C. Heinrich, “Spatially varying mixtures incorporating line processes for image segmentation,” J. Math Imaging Vis. 36, 91-110 (2010).
[CrossRef]

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

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

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

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

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

Opt. Eng. (1)

B. Aiazzi, L. Alparone, A. Barducci, S. Baronti, and I. Pippi, “Estimating noise and information of multispectral imagery,” Opt. Eng. 41, 656-668 (2002).
[CrossRef]

Opt. Express (2)

SIAM J. Optim. (1)

J. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, “Convergence properties of the Nelder-Mead simplex method in low dimensions,” SIAM J. Optim. 9, 112-147 (1998).
[CrossRef]

Other (5)

G. Sfikas, C. Heinrich, J. Zallat, C. Nikou, and N. Galatsanos, “Joint recovery and segmentation of polarimetric images using a compound mrf and mixture modeling,” in Proceedings of the IEEE International Conference on Image Processing (ICIP 09) (IEEE, 2009), pp. 3901-3904.
[CrossRef]

G. Birkhoff and S. MacLane, A Survey of Modern Algebra (McMillan, 1953).

G. McLachlan, Finite Mixture Models (Wiley-Interscience, 2000).
[CrossRef]

C. M. Bishop, Pattern Recognition and Machine Learning. (Springer, 2006).

K. Lange, Optimization (Springer, 2004).

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Figures (6)

Fig. 1
Fig. 1

Graphical model for the GMM-T Stokes image restoration/ segmentation model (see Section 3). Stokes vectors s n constitute the estimated restoration, produced by observations g n . Random variable sets u k n j , π k n constitute the smoothing prior set on the segmentation z k n . The Stokes vectors follow a truncated Gaussian distribution conditioned on the hidden variables z . Superscripts n [ 1 , N ] and j [ 1 , N ] denote pixel indices, subscript k [ 1 , K ] denotes kernel (segment) index, subscript d [ 1 , D ] indexes the neighborhood direction type. At most, Γ neighbors are defined for each site.

Fig. 2
Fig. 2

Graphical model for spatially variant GMM-IP Stokes image restoration/segmentation model (see Section 4). Stokes vectors s n constitute the estimated restoration, produced by observations g n . Random variable sets u k n j , π k n constitute the smoothing prior set on the segmentation z k n . The transformed Stokes image parameters λ n are assumed to be normally distributed conditioned on the hidden variables z . Superscripts n [ 1 , N ] and j [ 1 , N ] denote pixel indices, subscript k [ 1 , K ] denotes kernel (segment) index, subscript d [ 1 , D ] indexes the neighborhood direction type. At most, Γ neighbors are defined for each site.

Fig. 3
Fig. 3

Recovery results for simulated Stokes data. From left to right, each column shows the four channels of (a) the degraded image g (SNR of 10 dB ), (b) the original Stokes image s , and (c) the noncomplying to Stokes constraints PI recovery estimate. In (d) and (e) we see the restoration result s ^ obtained with our method, in (d) without the Stokes image transformation (Section 3) and in (e) with the Stokes image transformation (Section 4). The corresponding segmentations of the degraded image into K = 2 classes are shown at the top of columns (d) and (e). All Stokes channels share the same gray-level scale.

Fig. 4
Fig. 4

Recovery results for real Stokes data. From left to right, each column shows the four channels of (a) the observed image g and (b) the noncomplying to Stokes constraints PI recovery estimate. In (c) and (d) we see the restoration result s ^ obtained with our method, (c) without the Stokes image transformation (Section 3) and (d) with the Stokes image transformation (Section 4). The corresponding segmentations of the observed image into K = 6 classes are shown at the top of columns (c) and (d). Columns (b), (c), and (d) show, from top to bottom, s 1 , s 2 / s 1 , s 3 / s 1 , and s 4 / s 1 .

Fig. 5
Fig. 5

Intensity images acquired with four probing polarization states for the metallic car toy.

Fig. 6
Fig. 6

Processing results of a real image (metallic car toy). From top to bottom: DOP map computed from the PI solution (DOP values greater than 1 were truncated), non-physically admissible pixels (in white, 12% of the total number of pixels) corresponding to the PI solution, DOP map corresponding to the GMM-T model. The GMM-T DOP map is significantly smoother than the PI DOP map, which is what is expected given the considered object.

Tables (4)

Tables Icon

Table 1 Restoration Error Results for the Simulated Stokes Data of Fig. 3 a

Tables Icon

Table 2 Runtime in Seconds per Each EM Iteration, for the Proposed Algorithms a

Tables Icon

Table 3 Algorithm 1: Proposed Restoration/Segmentation Algorithm (Section 3, GMM-T)

Tables Icon

Table 4 Algorithm 2: Proposed Restoration/Segmentation Algorithm (Section 4, GMM-IP)

Equations (48)

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

Φ = [ E x E x * E x E y * E x * E y E y E y * ] = [ s 1 + s 2 s 3 i s 4 s 3 + i s 4 s 1 s 2 ] .
g n = H s n + ϵ n ,
s 1 n 0 , ( s 1 n ) 2 ( s 2 n ) 2 + ( s 3 n ) 2 + ( s 4 n ) 2 .
p ( g | s ; V ) = n = 1 N p ( g n | s n ; V ) , g n | s n ; V N ( H s n , V ) ,
z n | π n Mult ( π n ) .
p ( π ) C e ψ c ( π ) ,
π k n π k j S t ( 0 , β k d 2 , ν k d ) , n [ 1 , N ] , k [ 1 , K ] , d [ 1 , D ] , j γ d ( n ) .
π k n π k j N ( 0 , β k d 2 / u k n j ) , u k n j G ( ν k d / 2 , ν k d / 2 ) , n , k , d , j γ d ( n ) ,
p ( s | z ; μ , Σ ) = n = 1 N p ( s n | z n ; μ , Σ ) ,
p ( s n | z n ; μ , Σ ) k = 1 K [ I ( s n ) N ( s n ; μ k , Σ k ) ] z k n .
ln p ( g , s , π ; Ψ )
ln p ( g , s , π ; Ψ ) = ln p ( g | s , π ; Ψ ) + ln p ( s , π ; Ψ ) ,
Q ( ( s , π , Ψ ) , ( s , π , Ψ ) ( t ) ) = E z , u | g , ( s , π ) ( t ) ; Ψ ( t ) [ ln p ( g , z , u | s , π ; Ψ ) ] = u z ln p ( g , z , u | s , π ; Ψ ) p ( z , u | g , ( s , π ) ( t ) ; ( Ψ ) ( t ) ) d u ;
( s , π , Ψ ) ( t + 1 ) = arg max s , π , Ψ [ Q ( ( s , π , Ψ ) , ( s , π , Ψ ) ( t ) ) + ln p ( s , π ; Ψ ) ] .
Λ = [ λ 1 0 λ 3 + i λ 4 λ 2 ] .
1 2 Φ = Λ Λ H ,
λ n | z k n = 1 ; μ k , Σ k N ( μ k , Σ k ) .
ln p ( g , λ , π ; Ψ )
ISNR = 20 log 10 s g s s ^ ,
z k n ( t ) = π k n ( t ) N ( λ n ( t ) ; μ k ( t ) , Σ k ( t ) ) l = 1 K π l n ( t ) N ( λ n ( t ) ; μ l ( t ) , Σ l ( t ) ) ,
u k n j ( t ) = ζ k n j ( t ) / η k n j ( t ) ,
ln u k n j ( t ) = Ϝ ( ζ k n j ( t ) ) ln η k n j ( t ) ,
j γ d ( n ) ζ k n j ( t ) = 1 2 ( ν k d ( t ) + 1 ) , j γ d ( n ) η k n j ( t ) = 1 2 ( ν k d ( t ) + ( π k n ( t ) π k j ( t ) ) 2 β k d 2 ( t ) ) .
z k n ( t ) = π k n ( t ) ( Ξ ( μ k ( t ) , Σ k ( t ) ) ) 1 N ( s n ( t ) ; μ k ( t ) , Σ k ( t ) ) l = 1 K π l n ( t ) ( Ξ ( μ l ( t ) , Σ l ( t ) ) ) 1 N ( s n ( t ) ; μ l ( t ) , Σ l ( t ) ) ,
Q ( ( λ , π , Ψ ) , ( λ , π , Ψ ) ( t ) ) + ln p ( λ , π ; Ψ ) = E z , u | g , ( λ , π ) ( t ) ; Ψ ( t ) { ln p ( g , z , u | λ , π ; Ψ ) + ln p ( λ , π ; Ψ ) } = E z , u | g , ( λ , π ) ( t ) ; Ψ ( t ) { ln p ( g , z , u , λ , π ; Ψ ) } = E z | λ ( t ) , π ( t ) { ln p ( g | λ ; V ) } + E z | λ ( t ) , π ( t ) { ln p ( λ | z ; μ , Σ ) } + E z | λ ( t ) , π ( t ) { ln p ( z | π ) } + E u | π ( t ) { ln p ( π | u ; β ) } + E u | π ( t ) { ln p ( u ; ν ) } .
μ k ( t + 1 ) = n = 1 N z k n ( t ) λ n ( t ) n = 1 N z k n ( t ) ,
Σ k ( t + 1 ) = n = 1 N z k n ( t ) ( λ n ( t ) μ k ( t + 1 ) ) ( λ n ( t ) μ k ( t + 1 ) ) T n = 1 N z k n ( t ) ,
β k d 2 ( t + 1 ) = n = 1 N j γ d ( n ) u k n j ( t ) ( π k n ( t ) π k j ( t ) ) 2 n = 1 N | γ d ( n ) | ,
V ( t + 1 ) = ( 4 N ) 1 n = 1 N ( g n H s n ( t ) ) T ( g n H s n ( t ) ) I .
a k n ( π k n ( t + 1 ) ) 2 + b k n ( π k n ( t + 1 ) ) + c k n ( t + 1 ) = 0 ,
a k n = d = 1 D { β k d 2 ( t ) j γ d ( n ) u k n j ( t ) } ,
b k n = d = 1 D { β k d - 2 ( t ) j γ d ( n ) u k n j ( t ) π k j ( t ) } ,
c k n = 1 2 z k n ( t ) .
ln ( ν k d ( t + 1 ) / 2 ) Ϝ ( ν k d ( t + 1 ) / 2 ) + [ n = 1 N j γ d ( n ) ( ln u k n j ( t ) u k n j ( t ) ) n = 1 N | γ d ( n ) | ] + 1 = 0 ,
p ( s n | z n ; μ , Σ ) = k = 1 K [ Ξ ( μ k , Σ k ) 1 I ( s n ) N ( s n ; μ k , Σ k ) ] z k n ,
Ξ ( μ k , Σ k ) = R 4 I ( σ ) N ( σ ; μ k , Σ k ) d σ ,
Ξ ( μ k , Σ k ) 1 T τ = 1 T I ( σ ( τ ) ) ,
μ ˜ k ( t + 1 ) = n = 1 N z k n ( t ) s n n = 1 N z k n ( t ) ,
Σ ˜ k ( t + 1 ) = n = 1 N z k n ( t ) ( s n μ k ( t + 1 ) ) ( s n μ k ( t + 1 ) ) T n = 1 N z k n ( t ) ,
s n ( t + 1 ) T A n ( t + 1 ) s n ( t + 1 ) 2 b n ( t + 1 ) T s n ( t + 1 ) ,
A n ( t + 1 ) = H T V n ( t + 1 ) 1 H + k = 1 K z k n ( t ) Σ k ( t ) 1 ,
b n ( t + 1 ) = g n T V n ( t + 1 ) 1 H + k = 1 K z k n ( t ) μ k ( t ) T Σ k ( t ) 1 .
s n ( 0 ) = ( H T H ) 1 H T g n .
( s 1 n ) 2 = ( s 2 n ) 2 + ( s 3 n ) 2 + ( s 4 n ) 2 .
( g n H φ ( λ n ( t ) ) ) T V 1 ( g n H φ ( λ n ( t ) ) ) + k = 1 K ( λ n ( t ) μ k ) T Σ k 1 ( λ n ( t ) μ k ) z k n ,
h T Ω 1 h + λ T Ω 2 λ + ω 3 h + ω 4 λ ,
h H φ ( λ ) , Ω 1 V 1 , Ω 2 k = 1 K z k Σ k 1 , ω 3 2 g T V 1 , ω 4 2 k = 1 K z k μ k T Σ k 1 .
{ s 1 n = [ ( λ 1 n ) 2 + ( λ 2 n ) 2 + ( λ 3 n ) 2 + ( λ 4 n ) 2 ] , s 2 n = [ ( λ 1 n ) 2 ( λ 2 n ) 2 ( λ 3 n ) 2 ( λ 4 n ) 2 ] , s 3 n = 2 λ 1 n λ 3 n , s 4 n = 2 λ 1 n λ 4 n .

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