Abstract

This paper extends and refines previous work on clustering of polarization-encoded images. The polarization-encoded images used in this work are considered as multidimensional parametric images where a clustering scheme based on Markovian Bayesian inference is applied. Hidden Markov Chains Model (HMCM) and Hidden Hierarchical Markovian Model (HHMM) show to handle effectively Mueller images and give very good results for biological tissues (vegetal leaves). Pretreatments attempting to reduce the image dimensionality based on the Principal Component Analysis (PCA) turns out to be useless for Mueller matrix images.

© 2004 Optical Society of America

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References

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  1. J. Zallat, C. Collet, and Y. Takakura, “Clustering of polarization-encoded images,” Appl. Opt. 43, 283–292 (2004).
    [Crossref] [PubMed]
  2. J.M. Bueno and P. Artal, “Double-pass imaging polarimetry in the human eye,” Opt. Lett. 24, 64–66 (1999).
    [Crossref]
  3. P.Y. Gerligand, M.H. Smith, and R.A. Chipman, “Polarimetric images of a cone,” Opt. Express 4, 420–430 (1999). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-4-10-420
    [Crossref] [PubMed]
  4. G.D. Lewis, D.L. Jordan, and P.J. Roberts, “Backscattering target detection in a turbid medium by polarization discrimination,” Appl. Opt. 38, 3937–3944 (1999).
    [Crossref]
  5. J.-N. Provost, C. Collet, P. Rostaing, P. Pérez, and P. Bouthemy, “Hierarchical Markovian Segmentation of Multispectral Images for the Reconstruction of Water Depth Maps,” Computer Vision and Image Understanding 93, 155–174 (2004).
    [Crossref]
  6. C. Collet, M. Louys, C. Bot, and A. Oberto, “Markov Model for Multispectral Image analysis : application to Small Magellanic Cloud segmentation,” in International Conference on Image Processing ICIP’03.2003. Barcelona, Spain.
  7. P.A. Devijver, “Baum’s forward-backward algorithm revisited,” Pattern Recognition Lett. 39, 369–373 (1985).
    [Crossref]
  8. N. Giordana and W. Pieczynski, “Estimation of generalized multisensor hidden Markov chains and unsupervised image segmentation,” IEEE Transactions on Pattern Analysis and Machine Intelligence 19, 465–475 (1997).
    [Crossref]
  9. J. M. Laferté, P. Pérez, and F. Heitz, “Discrete Markov Image Modeling and Inference on the Quad-tree,” IEEE Transactions on Image Processing 9, 390–404 (2000).
    [Crossref]
  10. C. H. Chen, L.F. Pau, and P.S.P. Wang, Handbook of Pattern Recognition and Computer Vision. WORLD SCIENTIFIC. 1044 (2000.)
  11. A. P. Dempster, A.P., N.M. Laird, and D.B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” Royal Statistical Society, 1–381976.
  12. R. O. Duda, P.E. Hart, and D.G. Stork, Pattern Classification, 2nd Edition Ed. Cloth. 680 (2000).
  13. Movie of Mueller images after PCA transform. 2004. ftp://picabia.u-strasbg.fr/pub/www/collet/Publis/Animations/acp.gif
  14. S. D. Baker, Unsupervised pattern recognition, PhD Thesis Antwerpen University (2002). http://www.ruca.ua.ac.be/visielab/theses/debacker/SteveThesis.pdf
  15. A. Hyvärinen, J. Karhunen, and E. Oja, Independent Component Analysis, John Wiley & Sons (2001).
    [Crossref]
  16. Movie of HMCM segmentation. 2004. ftp://picabia.u-strasbg.fr/pub/www/collet/Publis/Animations/Iteration_map.gif
  17. Movie of HMCM segmentation map with different number of classes. 2004. ftp://picabia.u-strasbg.fr/pub/www/collet/Publis/Animations/3-6classes_resample.gif
  18. Movie of HHMM segmentation. 2004. ftp://picabia.u-strasbg.fr/pub/www/collet/Publis/Animations/HHMM_animation.gif
  19. S. R. Cloude and E. Pottier, “Concept of polarization entropy in optical scattering,” Opt. Eng. 61599–610 (1995).
    [Crossref]
  20. S. Y. Lu and R.A. Chipman, “Interpretation of Mueller matrices based on polar decomposition,” J. Opt. Soc. Am. A 131–8 (1996).
    [Crossref]

2004 (2)

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

J.-N. Provost, C. Collet, P. Rostaing, P. Pérez, and P. Bouthemy, “Hierarchical Markovian Segmentation of Multispectral Images for the Reconstruction of Water Depth Maps,” Computer Vision and Image Understanding 93, 155–174 (2004).
[Crossref]

2000 (1)

J. M. Laferté, P. Pérez, and F. Heitz, “Discrete Markov Image Modeling and Inference on the Quad-tree,” IEEE Transactions on Image Processing 9, 390–404 (2000).
[Crossref]

1999 (3)

1997 (1)

N. Giordana and W. Pieczynski, “Estimation of generalized multisensor hidden Markov chains and unsupervised image segmentation,” IEEE Transactions on Pattern Analysis and Machine Intelligence 19, 465–475 (1997).
[Crossref]

1996 (1)

S. Y. Lu and R.A. Chipman, “Interpretation of Mueller matrices based on polar decomposition,” J. Opt. Soc. Am. A 131–8 (1996).
[Crossref]

1995 (1)

S. R. Cloude and E. Pottier, “Concept of polarization entropy in optical scattering,” Opt. Eng. 61599–610 (1995).
[Crossref]

1985 (1)

P.A. Devijver, “Baum’s forward-backward algorithm revisited,” Pattern Recognition Lett. 39, 369–373 (1985).
[Crossref]

A.P.,

A. P. Dempster, A.P., N.M. Laird, and D.B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” Royal Statistical Society, 1–381976.

Artal, P.

Baker, S. D.

S. D. Baker, Unsupervised pattern recognition, PhD Thesis Antwerpen University (2002). http://www.ruca.ua.ac.be/visielab/theses/debacker/SteveThesis.pdf

Bot, C.

C. Collet, M. Louys, C. Bot, and A. Oberto, “Markov Model for Multispectral Image analysis : application to Small Magellanic Cloud segmentation,” in International Conference on Image Processing ICIP’03.2003. Barcelona, Spain.

Bouthemy, P.

J.-N. Provost, C. Collet, P. Rostaing, P. Pérez, and P. Bouthemy, “Hierarchical Markovian Segmentation of Multispectral Images for the Reconstruction of Water Depth Maps,” Computer Vision and Image Understanding 93, 155–174 (2004).
[Crossref]

Bueno, J.M.

Chen, C. H.

C. H. Chen, L.F. Pau, and P.S.P. Wang, Handbook of Pattern Recognition and Computer Vision. WORLD SCIENTIFIC. 1044 (2000.)

Chipman, R.A.

Cloude, S. R.

S. R. Cloude and E. Pottier, “Concept of polarization entropy in optical scattering,” Opt. Eng. 61599–610 (1995).
[Crossref]

Collet, C.

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

J.-N. Provost, C. Collet, P. Rostaing, P. Pérez, and P. Bouthemy, “Hierarchical Markovian Segmentation of Multispectral Images for the Reconstruction of Water Depth Maps,” Computer Vision and Image Understanding 93, 155–174 (2004).
[Crossref]

C. Collet, M. Louys, C. Bot, and A. Oberto, “Markov Model for Multispectral Image analysis : application to Small Magellanic Cloud segmentation,” in International Conference on Image Processing ICIP’03.2003. Barcelona, Spain.

Dempster, A. P.

A. P. Dempster, A.P., N.M. Laird, and D.B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” Royal Statistical Society, 1–381976.

Devijver, P.A.

P.A. Devijver, “Baum’s forward-backward algorithm revisited,” Pattern Recognition Lett. 39, 369–373 (1985).
[Crossref]

Duda, R. O.

R. O. Duda, P.E. Hart, and D.G. Stork, Pattern Classification, 2nd Edition Ed. Cloth. 680 (2000).

Gerligand, P.Y.

Giordana, N.

N. Giordana and W. Pieczynski, “Estimation of generalized multisensor hidden Markov chains and unsupervised image segmentation,” IEEE Transactions on Pattern Analysis and Machine Intelligence 19, 465–475 (1997).
[Crossref]

Hart, P.E.

R. O. Duda, P.E. Hart, and D.G. Stork, Pattern Classification, 2nd Edition Ed. Cloth. 680 (2000).

Heitz, F.

J. M. Laferté, P. Pérez, and F. Heitz, “Discrete Markov Image Modeling and Inference on the Quad-tree,” IEEE Transactions on Image Processing 9, 390–404 (2000).
[Crossref]

Hyvärinen, A.

A. Hyvärinen, J. Karhunen, and E. Oja, Independent Component Analysis, John Wiley & Sons (2001).
[Crossref]

Jordan, D.L.

Karhunen, J.

A. Hyvärinen, J. Karhunen, and E. Oja, Independent Component Analysis, John Wiley & Sons (2001).
[Crossref]

Laferté, J. M.

J. M. Laferté, P. Pérez, and F. Heitz, “Discrete Markov Image Modeling and Inference on the Quad-tree,” IEEE Transactions on Image Processing 9, 390–404 (2000).
[Crossref]

Laird, N.M.

A. P. Dempster, A.P., N.M. Laird, and D.B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” Royal Statistical Society, 1–381976.

Lewis, G.D.

Louys, M.

C. Collet, M. Louys, C. Bot, and A. Oberto, “Markov Model for Multispectral Image analysis : application to Small Magellanic Cloud segmentation,” in International Conference on Image Processing ICIP’03.2003. Barcelona, Spain.

Lu, S. Y.

S. Y. Lu and R.A. Chipman, “Interpretation of Mueller matrices based on polar decomposition,” J. Opt. Soc. Am. A 131–8 (1996).
[Crossref]

Oberto, A.

C. Collet, M. Louys, C. Bot, and A. Oberto, “Markov Model for Multispectral Image analysis : application to Small Magellanic Cloud segmentation,” in International Conference on Image Processing ICIP’03.2003. Barcelona, Spain.

Oja, E.

A. Hyvärinen, J. Karhunen, and E. Oja, Independent Component Analysis, John Wiley & Sons (2001).
[Crossref]

Pau, L.F.

C. H. Chen, L.F. Pau, and P.S.P. Wang, Handbook of Pattern Recognition and Computer Vision. WORLD SCIENTIFIC. 1044 (2000.)

Pérez, P.

J.-N. Provost, C. Collet, P. Rostaing, P. Pérez, and P. Bouthemy, “Hierarchical Markovian Segmentation of Multispectral Images for the Reconstruction of Water Depth Maps,” Computer Vision and Image Understanding 93, 155–174 (2004).
[Crossref]

J. M. Laferté, P. Pérez, and F. Heitz, “Discrete Markov Image Modeling and Inference on the Quad-tree,” IEEE Transactions on Image Processing 9, 390–404 (2000).
[Crossref]

Pieczynski, W.

N. Giordana and W. Pieczynski, “Estimation of generalized multisensor hidden Markov chains and unsupervised image segmentation,” IEEE Transactions on Pattern Analysis and Machine Intelligence 19, 465–475 (1997).
[Crossref]

Pottier, E.

S. R. Cloude and E. Pottier, “Concept of polarization entropy in optical scattering,” Opt. Eng. 61599–610 (1995).
[Crossref]

Provost, J.-N.

J.-N. Provost, C. Collet, P. Rostaing, P. Pérez, and P. Bouthemy, “Hierarchical Markovian Segmentation of Multispectral Images for the Reconstruction of Water Depth Maps,” Computer Vision and Image Understanding 93, 155–174 (2004).
[Crossref]

Roberts, P.J.

Rostaing, P.

J.-N. Provost, C. Collet, P. Rostaing, P. Pérez, and P. Bouthemy, “Hierarchical Markovian Segmentation of Multispectral Images for the Reconstruction of Water Depth Maps,” Computer Vision and Image Understanding 93, 155–174 (2004).
[Crossref]

Rubin, D.B.

A. P. Dempster, A.P., N.M. Laird, and D.B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” Royal Statistical Society, 1–381976.

Smith, M.H.

Stork, D.G.

R. O. Duda, P.E. Hart, and D.G. Stork, Pattern Classification, 2nd Edition Ed. Cloth. 680 (2000).

Takakura, Y.

Wang, P.S.P.

C. H. Chen, L.F. Pau, and P.S.P. Wang, Handbook of Pattern Recognition and Computer Vision. WORLD SCIENTIFIC. 1044 (2000.)

Zallat, J.

Appl. Opt. (2)

Computer Vision and Image Understanding (1)

J.-N. Provost, C. Collet, P. Rostaing, P. Pérez, and P. Bouthemy, “Hierarchical Markovian Segmentation of Multispectral Images for the Reconstruction of Water Depth Maps,” Computer Vision and Image Understanding 93, 155–174 (2004).
[Crossref]

IEEE Transactions on Image Processing (1)

J. M. Laferté, P. Pérez, and F. Heitz, “Discrete Markov Image Modeling and Inference on the Quad-tree,” IEEE Transactions on Image Processing 9, 390–404 (2000).
[Crossref]

IEEE Transactions on Pattern Analysis and Machine Intelligence (1)

N. Giordana and W. Pieczynski, “Estimation of generalized multisensor hidden Markov chains and unsupervised image segmentation,” IEEE Transactions on Pattern Analysis and Machine Intelligence 19, 465–475 (1997).
[Crossref]

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

S. Y. Lu and R.A. Chipman, “Interpretation of Mueller matrices based on polar decomposition,” J. Opt. Soc. Am. A 131–8 (1996).
[Crossref]

Opt. Eng. (1)

S. R. Cloude and E. Pottier, “Concept of polarization entropy in optical scattering,” Opt. Eng. 61599–610 (1995).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Pattern Recognition Lett. (1)

P.A. Devijver, “Baum’s forward-backward algorithm revisited,” Pattern Recognition Lett. 39, 369–373 (1985).
[Crossref]

Other (10)

C. Collet, M. Louys, C. Bot, and A. Oberto, “Markov Model for Multispectral Image analysis : application to Small Magellanic Cloud segmentation,” in International Conference on Image Processing ICIP’03.2003. Barcelona, Spain.

C. H. Chen, L.F. Pau, and P.S.P. Wang, Handbook of Pattern Recognition and Computer Vision. WORLD SCIENTIFIC. 1044 (2000.)

A. P. Dempster, A.P., N.M. Laird, and D.B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” Royal Statistical Society, 1–381976.

R. O. Duda, P.E. Hart, and D.G. Stork, Pattern Classification, 2nd Edition Ed. Cloth. 680 (2000).

Movie of Mueller images after PCA transform. 2004. ftp://picabia.u-strasbg.fr/pub/www/collet/Publis/Animations/acp.gif

S. D. Baker, Unsupervised pattern recognition, PhD Thesis Antwerpen University (2002). http://www.ruca.ua.ac.be/visielab/theses/debacker/SteveThesis.pdf

A. Hyvärinen, J. Karhunen, and E. Oja, Independent Component Analysis, John Wiley & Sons (2001).
[Crossref]

Movie of HMCM segmentation. 2004. ftp://picabia.u-strasbg.fr/pub/www/collet/Publis/Animations/Iteration_map.gif

Movie of HMCM segmentation map with different number of classes. 2004. ftp://picabia.u-strasbg.fr/pub/www/collet/Publis/Animations/3-6classes_resample.gif

Movie of HHMM segmentation. 2004. ftp://picabia.u-strasbg.fr/pub/www/collet/Publis/Animations/HHMM_animation.gif

Supplementary Material (2)

» Media 1: GIF (1584 KB)     
» Media 2: GIF (1075 KB)     

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Figures (6)

Fig. 1.
Fig. 1.

Schematic of the Mueller matrix imaging polarimeter used in this study.

Fig. 2.
Fig. 2.

(a) Markov chain model and Hilbert Peano paths for different image sizes (22, 42, 82 and 162 pixels). The image becomes a vector after Hilbert-Peano scan. Markovian prior model describes the transitions within the chain between labels Xn and Xn +1 whereas the data-driven parameter links X n+1 and Y n+1. (b) Markovian quadtree with inter-scale transition probabilities aij (s- stands for the father of site s in the tree). Data-driven probability density function between observations (white circle) and labels (black circle) equals fi (l)=p(ys =l/xs =ωi )

Fig. 3.
Fig. 3.

(a) m 00 Mueller element image. This element image corresponds to a conventional intensity image with which results must be compared. (b) HMCM analysis with 6 classes

Fig. 4.
Fig. 4.

. (a) (1.54 Mb) movie of raw Mueller images (16 parameters for each pixel: (mij; i,j=0…3). (b) (1.04 Mb) Sequence of HMCM maps from initialization (based on K-mean algorithm) up to label map: convergence is performed in 6 iterations, taking less than 1 mn on a PC Pentium IV, 2 GHz, 4 Gb RAM.. Each class is displayed with a random gray level.

Fig. 5.
Fig. 5.

(a) Segmentation map after PCA transform (b) HHMM map : convergence is performed in 6 iterations, taking less than 1 mn on a PC Pentium IV, 2 GHz, 4 Gb RAM. One observes weak blocky effects. This label map is similar to HMCM map obtained on Fig. 4(b).

Fig. 6.
Fig. 6.

Eigenvalues of O(M N ) and O(M T )

Tables (1)

Tables Icon

Table 1. Main interaction mechanism + isotropic depolarizer results

Equations (12)

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

I ij = 1 2 ( 1 cos 2 2 α 2 k 0.5 sin 2 4 α 2 k sin 2 α 2 k ) M ij [ 1 cos 2 2 α 1 l 0.5 sin 2 4 α 1 l sin 2 α 1 l ]
M ij = A 1 I ij P 1
A = [ 1 cos 2 2 α 2 1 0.5 sin 2 α 2 1 sin 2 α 2 1 1 cos 2 2 α 2 2 0.5 sin 2 α 2 2 sin 2 α 2 2 1 cos 2 2 α 2 3 0.5 sin 2 α 2 3 sin 2 α 2 3 1 cos 2 2 α 2 4 0.5 sin 2 α 2 4 sin 2 α 2 4 ]
P = [ 1 1 1 1 cos 2 2 α 1 2 cos 2 2 α 1 2 cos 2 2 α 1 2 cos 2 2 α 1 2 0.5 sin 2 α 1 3 0.5 sin 2 α 1 3 0.5 sin 2 α 1 3 0.5 sin 2 α 1 3 sin 2 α 1 4 sin 2 α 1 4 sin 2 α 1 4 sin 2 α 1 4 ]
H = i = 0 3 h i log 4 h i ; h i = λ i j = 0 , 3 λ j
M = M dep · M R · M D
p X ( X ) = 1 Z exp [ c C Ψ c ( x c ) ]
max X p ( X , Y ) max X p ( X Y ) = 1 Z Y exp [ ( c C Ψ c ( x c ) s S Log p ( y s x s ) ) ]
M N = ( 1.0000 0.0227 0.0031 0.0028 0.0077 0.2066 0.0038 0.0096 0.0009 0.0121 0.2225 0.0024 0.0035 0.0118 0.0082 0.1306 )
M T = ( 1.0000 0.0269 0.0021 0.0018 0.0101 0.3236 0.0087 0.0023 0.0008 0.0024 0.3276 0.0009 0.0026 0.0023 0.0029 0.2754 )
M 0 = O 1 ( ( λ 0 d 2 ) v 0 v 0 t * )
d = 2 3 ( λ 1 + λ 2 + λ 3 )

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