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]
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    [CrossRef]
  3. P.Y. Gerligand, M.H. Smith, and R.A. Chipman, �??Polarimetric images of a cone,�?? Opt. Express 4, 420-430 (1999). <a href= "http://www.opticsexpress.org/abstract.cfm?URI=OPEX-4-10-420"> http://www.opticsexpress.org/abstract.cfm?URI=OPEX-4-10-420</a>.
    [CrossRef] [PubMed]
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  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. Collet, C., 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]
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    [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, 1-38 1976.
  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. <a href="ftp://picabia.u-strasbg.fr/pub/www/collet/Publis/Animations/acp.gif">ftp://picabia.u-strasbg.fr/pub/www/collet/Publis/Animations/acp.gif</a>
  14. S. D. Baker, Unsupervised pattern recognition, PhD Thesis Antwerpen University (2002). <a href= "http://www.ruca.ua.ac.be/visielab/theses/debacker/SteveThesis.pdf"> http://www.ruca.ua.ac.be/visielab/theses/debacker/SteveThesis.pdf</a>
  15. A. Hyvärinen, J. Karhunen, and E. Oja, Independent Component Analysis, John Wiley & Sons (2001).
    [CrossRef]
  16. Movie of HMCM segmentation. 2004. <a href= "ftp://picabia.u-strasbg.fr/pub/www/collet/Publis/Animations/Iteration_map.gif"> ftp://picabia.u-strasbg.fr/pub/www/collet/Publis/Animations/Iteration_map.gif</a>
  17. Movie of HMCM segmentation map with different number of classes. 2004. <a href="ftp://picabia.u-strasbg.fr/pub/www/collet/Publis/Animations/3-classes_resample.gif"> ftp://picabia.u-strasbg.fr/pub/www/collet/Publis/Animations/3-classes_resample.gif</a>
  18. Movie of HHMM segmentation. 2004. <a href= "ftp://picabia.u-strasbg.fr/pub/www/collet/Publis/Animations/HHMM_animation.gif"> ftp://picabia.u-strasbg.fr/pub/www/collet/Publis/Animations/HHMM_animation.gif</a>
  19. S. R. Cloude and E. Pottier, �??Concept of polarization entropy in optical scattering,�?? Opt. Eng. 6 1599-610 (1995).
    [CrossRef]

Appl. Opt.

Computer Vision and Image Understanding

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]

ICIP 2003

Collet, C., 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.

IEEE Tran on Pattern Anal and Mach Intel

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]

IEEE Transactions on Image Processing

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]

Opt. Eng.

S. R. Cloude and E. Pottier, �??Concept of polarization entropy in optical scattering,�?? Opt. Eng. 6 1599-610 (1995).
[CrossRef]

Opt. Express

Opt. Lett.

Pattern Recognition Lett.

P.A. Devijver , �??Baum's forward-backward algorithm revisited,�?? Pattern Recognition Lett. 39, 369-373 (1985).
[CrossRef]

Royal Statistical Society

A. P. Dempster, A.P., N.M. Laird, and D.B. Rubin, �??Maximum likelihood from incomplete data via the EM algorithm,�?? Royal Statistical Society, 1-38 1976.

Other

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. <a href="ftp://picabia.u-strasbg.fr/pub/www/collet/Publis/Animations/acp.gif">ftp://picabia.u-strasbg.fr/pub/www/collet/Publis/Animations/acp.gif</a>

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

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

Movie of HMCM segmentation. 2004. <a href= "ftp://picabia.u-strasbg.fr/pub/www/collet/Publis/Animations/Iteration_map.gif"> ftp://picabia.u-strasbg.fr/pub/www/collet/Publis/Animations/Iteration_map.gif</a>

Movie of HMCM segmentation map with different number of classes. 2004. <a href="ftp://picabia.u-strasbg.fr/pub/www/collet/Publis/Animations/3-classes_resample.gif"> ftp://picabia.u-strasbg.fr/pub/www/collet/Publis/Animations/3-classes_resample.gif</a>

Movie of HHMM segmentation. 2004. <a href= "ftp://picabia.u-strasbg.fr/pub/www/collet/Publis/Animations/HHMM_animation.gif"> ftp://picabia.u-strasbg.fr/pub/www/collet/Publis/Animations/HHMM_animation.gif</a>

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

Supplementary Material (2)

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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)

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Table 1. Main interaction mechanism + isotropic depolarizer results

Equations (12)

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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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