Abstract

Polarization-encoded imaging consists of the distributed measurements of polarization parameters for each pixel of an image. We address clustering of multidimensional polarization-encoded images. The spatial coherence of polarization information is considered. Two methods of analysis are proposed: polarization contrast enhancement and a more-sophisticated image-processing algorithm based on a Markovian model. The proposed algorithms are applied and validated with two different Mueller images acquired by a fully polarimetric imaging system.

© 2004 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  32. M. Mignotte, C. Collet, P. Pérez, P. Bouthemy, “Three-class Markovian segmentation of high resolution sonar images,” J. Computer Vision Image Underst. 76, 191–204 (1999).
    [CrossRef]
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    [CrossRef]
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  36. J.-M. Laferté, P. Pérez, F. Heitz, “Discrete Markov image modeling and inference on the quad-tree,” IEEE Trans. Image Process. 9, 390–404 (2000).
    [CrossRef]
  37. P. Rostaing, P. Pérez, J.-N. Provost, C. Collet, P. Bouthemy, “Multispectral spot images analysis using generalized Gaussian modeling: application to water depth mapping,” Comput. Vis. Image Underst. (to be published).
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    [CrossRef]

2001

F. Goudail, P. Réfrégier, “Statistical algorithms for processing polarimetric images in coherent light,” Trait. Signal 18(5–6), 297–319 (2001).

2000

P. Pérez, A. Chardin, J.-M. Laferté, “Noniterative manipulation of discrete energy-based models for image analysis,” Pattern Recogn. 33, 573–586 (2000).
[CrossRef]

J.-M. Laferté, P. Pérez, F. Heitz, “Discrete Markov image modeling and inference on the quad-tree,” IEEE Trans. Image Process. 9, 390–404 (2000).
[CrossRef]

1999

1998

1997

N. Giordana, W. Pieczynski, “Estimation of generalized multisensor hidden Markov chains and unsupervised image segmentation,” IEEE Trans. Pattern Anal. Mach. Intell. 19, 465–475 (1997).
[CrossRef]

1996

P. Pérez, F. Heitz, “Restriction of a Markov random field on a graph and multiresolution statistical image modeling,” IEEE Trans. Inf. Theory 42, 180–190 (1996).
[CrossRef]

Z. Kato, M. Berthod, J. Zérubia, “A hierarchical Markov random field model and multitemperature annealing for parallel image classification,” Graph. Models Image Process. 58, 18–37 (1996).
[CrossRef]

S.-Y. Lu, R. A. Chipman, “Interpretation of Mueller matrices based on polar decomposition,” J. Opt. Soc. Am. A 13, 1106–1113 (1996).
[CrossRef]

1994

D. G. M. Anderson, R. Barakat, “Necessary and sufficient conditions for a Mueller matrix to be derivable from a Jones matrix,” J. Opt. Soc. Am. A 11, 2305–2319 (1994).
[CrossRef]

C. A. Bouman, M. Shapiro, “A multiscale random field model for Bayesian image segmentation,” IEEE Trans. Image Process. 3, 162–177 (1994).
[CrossRef] [PubMed]

R. Sridhar, R. Simon, “Normal form for Mueller matrices in polarization optics,” J. Mod. Opt. 41, 1903–1915 (1994).
[CrossRef]

1993

C. R. Givens, A. B. Kostinski, “A simple necessary and sufficient condition on physically realizable Mueller matrices,” J. Mod. Opt. 40, 471–481 (1993).
[CrossRef]

C. V. M. van der Mee, “An eigenvalue criterion for matrices transforming Stokes parameters,” J. Math. Phys. 34, 5072–5088 (1993).
[CrossRef]

M. R. Luettgen, W. C. Karl, A. S. Willsky, R. Tenney, “Multiscale representation of Markov random fields,” IEEE Trans. Signal Process. 41, 3377–3395 (1993).
[CrossRef]

A. B. Kostinski, C. R. Givens, J. M. Kwiatkowski, “Constraints on Mueller matrices of polarization optics,” Appl. Opt. 32, 1646–1651 (1993).
[CrossRef] [PubMed]

1992

M. S. Kumar, R. Simon, “Characterization of Mueller matrices in polarization optics,” Opt. Commun. 88, 464–470 (1992).
[CrossRef]

W. Pieczynski, “Statistical image segmentation,” Mach. Graph. Vision 1, 261–268 (1992).

Z.-F. Xing, “On the deterministic and non-deterministic Mueller matrix,” J. Mod. Opt. 39, 461–484 (1992).
[CrossRef]

C. V. M. van der Mee, J. W. Hovenier, “Structure of matrices transforming Stokes parameters,” J. Math. Phys. 33, 3574–3584 (1992).
[CrossRef]

1986

S. R. Cloude, “Group theory and polarisation algebra,” Optik (Stuttgart) 75, 26–36 (1986).

1984

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

1981

1979

1978

Anderson, D. G. M.

Artal, P.

Ax, G. R.

M. H. Smith, J. D. Howe, J. B. Woodruff, M. A. Miller, G. R. Ax, T. E. Petty, E. A. Sornsin, “Multispectral infrared Stokes imaging polarimeter,” in Polarization: Measurement, Analysis, and Remote Sensing, D. B. Chenault, M. J. Duggin, W. G. Egan, D. H. Goldstein, eds., Proc. SPIE3754, 137–143.

Azzam, R. M. A.

Barakat, R.

Berthod, M.

Z. Kato, M. Berthod, J. Zérubia, “A hierarchical Markov random field model and multitemperature annealing for parallel image classification,” Graph. Models Image Process. 58, 18–37 (1996).
[CrossRef]

Bouman, C. A.

C. A. Bouman, M. Shapiro, “A multiscale random field model for Bayesian image segmentation,” IEEE Trans. Image Process. 3, 162–177 (1994).
[CrossRef] [PubMed]

Bouthemy, P.

M. Mignotte, C. Collet, P. Pérez, P. Bouthemy, “Three-class Markovian segmentation of high resolution sonar images,” J. Computer Vision Image Underst. 76, 191–204 (1999).
[CrossRef]

P. Rostaing, P. Pérez, J.-N. Provost, C. Collet, P. Bouthemy, “Multispectral spot images analysis using generalized Gaussian modeling: application to water depth mapping,” Comput. Vis. Image Underst. (to be published).

Bueno, J. M.

Chardin, A.

P. Pérez, A. Chardin, J.-M. Laferté, “Noniterative manipulation of discrete energy-based models for image analysis,” Pattern Recogn. 33, 573–586 (2000).
[CrossRef]

Chipman, R. A.

P. Y. Gerligand, M. H. Smith, R. A. Chipman, “Polarimetric images of a cone,” Opt. Express 4, 420–430 (1999), http://www.opticsexpress.org .
[CrossRef] [PubMed]

S.-Y. Lu, R. A. Chipman, “Mueller matrices and the degree of polarization,” Opt. Commun. 146, 11–14 (1998).
[CrossRef]

S.-Y. Lu, R. A. Chipman, “Interpretation of Mueller matrices based on polar decomposition,” J. Opt. Soc. Am. A 13, 1106–1113 (1996).
[CrossRef]

R. A. Chipman, “Polarimetry,” in Handbook of Optics, M. Bass, ed. (McGraw-Hill, New York, 1993), pp. 22.1–22.33.

Cloude, S. R.

S. R. Cloude, “Group theory and polarisation algebra,” Optik (Stuttgart) 75, 26–36 (1986).

Collet, C.

M. Mignotte, C. Collet, P. Pérez, P. Bouthemy, “Three-class Markovian segmentation of high resolution sonar images,” J. Computer Vision Image Underst. 76, 191–204 (1999).
[CrossRef]

P. Rostaing, J.-N. Provost, C. Collet, “Unsupervised multispectral image segmentation using generalized Gaussian noise model,” in Proceedings of the International Workshop EMMCVPR’99, Vol. 1654 of Springer Lecture Notes in Computer Science (Springer-Verlag, Berlin, 1999), pp. 141–156.

P. Rostaing, P. Pérez, J.-N. Provost, C. Collet, P. Bouthemy, “Multispectral spot images analysis using generalized Gaussian modeling: application to water depth mapping,” Comput. Vis. Image Underst. (to be published).

del Toro Iniesta, J. C.

Duda, R. O.

R. O. Duda, P. E. Hart, Pattern Classification and Scene Analysis (Wiley Interscience, New York, 1973).

Fry, E. S.

Geman, D.

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

Geman, S.

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

Gerligand, P. Y.

Giordana, N.

N. Giordana, W. Pieczynski, “Estimation of generalized multisensor hidden Markov chains and unsupervised image segmentation,” IEEE Trans. Pattern Anal. Mach. Intell. 19, 465–475 (1997).
[CrossRef]

Givens, C. R.

A. B. Kostinski, C. R. Givens, J. M. Kwiatkowski, “Constraints on Mueller matrices of polarization optics,” Appl. Opt. 32, 1646–1651 (1993).
[CrossRef] [PubMed]

C. R. Givens, A. B. Kostinski, “A simple necessary and sufficient condition on physically realizable Mueller matrices,” J. Mod. Opt. 40, 471–481 (1993).
[CrossRef]

Goudail, F.

F. Goudail, P. Réfrégier, “Statistical algorithms for processing polarimetric images in coherent light,” Trait. Signal 18(5–6), 297–319 (2001).

Graffigne, C.

C. Graffigne, F. Heitz, Pérez, F. Prêteux, M. Sigelle, J. Zerubia, “Hierarchical Markov random field models applied to image analysis: a review,” in Neural, Morphological, and Stochastic Methods in Image and Signal Processing, E. R. Dougherty, F. J. Preteux, S. S. Shen, eds., Proc. SPIE2568, 2–17 (1995).
[CrossRef]

Hart, P. E.

R. O. Duda, P. E. Hart, Pattern Classification and Scene Analysis (Wiley Interscience, New York, 1973).

Heitz, F.

J.-M. Laferté, P. Pérez, F. Heitz, “Discrete Markov image modeling and inference on the quad-tree,” IEEE Trans. Image Process. 9, 390–404 (2000).
[CrossRef]

P. Pérez, F. Heitz, “Restriction of a Markov random field on a graph and multiresolution statistical image modeling,” IEEE Trans. Inf. Theory 42, 180–190 (1996).
[CrossRef]

C. Graffigne, F. Heitz, Pérez, F. Prêteux, M. Sigelle, J. Zerubia, “Hierarchical Markov random field models applied to image analysis: a review,” in Neural, Morphological, and Stochastic Methods in Image and Signal Processing, E. R. Dougherty, F. J. Preteux, S. S. Shen, eds., Proc. SPIE2568, 2–17 (1995).
[CrossRef]

Hovenier, J. W.

C. V. M. van der Mee, J. W. Hovenier, “Structure of matrices transforming Stokes parameters,” J. Math. Phys. 33, 3574–3584 (1992).
[CrossRef]

Howe, J. D.

M. H. Smith, J. D. Howe, J. B. Woodruff, M. A. Miller, G. R. Ax, T. E. Petty, E. A. Sornsin, “Multispectral infrared Stokes imaging polarimeter,” in Polarization: Measurement, Analysis, and Remote Sensing, D. B. Chenault, M. J. Duggin, W. G. Egan, D. H. Goldstein, eds., Proc. SPIE3754, 137–143.

Howell, B. J.

Jordan, D. L.

Karl, W. C.

M. R. Luettgen, W. C. Karl, A. S. Willsky, R. Tenney, “Multiscale representation of Markov random fields,” IEEE Trans. Signal Process. 41, 3377–3395 (1993).
[CrossRef]

Kato, Z.

Z. Kato, M. Berthod, J. Zérubia, “A hierarchical Markov random field model and multitemperature annealing for parallel image classification,” Graph. Models Image Process. 58, 18–37 (1996).
[CrossRef]

Kattawar, G. W.

Kostinski, A. B.

A. B. Kostinski, C. R. Givens, J. M. Kwiatkowski, “Constraints on Mueller matrices of polarization optics,” Appl. Opt. 32, 1646–1651 (1993).
[CrossRef] [PubMed]

C. R. Givens, A. B. Kostinski, “A simple necessary and sufficient condition on physically realizable Mueller matrices,” J. Mod. Opt. 40, 471–481 (1993).
[CrossRef]

Kumar, M. S.

M. S. Kumar, R. Simon, “Characterization of Mueller matrices in polarization optics,” Opt. Commun. 88, 464–470 (1992).
[CrossRef]

Kwiatkowski, J. M.

Laferté, J.-M.

P. Pérez, A. Chardin, J.-M. Laferté, “Noniterative manipulation of discrete energy-based models for image analysis,” Pattern Recogn. 33, 573–586 (2000).
[CrossRef]

J.-M. Laferté, P. Pérez, F. Heitz, “Discrete Markov image modeling and inference on the quad-tree,” IEEE Trans. Image Process. 9, 390–404 (2000).
[CrossRef]

Landi degl’ Innocenti, E.

Lewis, G. D.

Lu, S.-Y.

S.-Y. Lu, R. A. Chipman, “Mueller matrices and the degree of polarization,” Opt. Commun. 146, 11–14 (1998).
[CrossRef]

S.-Y. Lu, R. A. Chipman, “Interpretation of Mueller matrices based on polar decomposition,” J. Opt. Soc. Am. A 13, 1106–1113 (1996).
[CrossRef]

Luettgen, M. R.

M. R. Luettgen, W. C. Karl, A. S. Willsky, R. Tenney, “Multiscale representation of Markov random fields,” IEEE Trans. Signal Process. 41, 3377–3395 (1993).
[CrossRef]

Mignotte, M.

M. Mignotte, C. Collet, P. Pérez, P. Bouthemy, “Three-class Markovian segmentation of high resolution sonar images,” J. Computer Vision Image Underst. 76, 191–204 (1999).
[CrossRef]

Miller, M. A.

M. H. Smith, J. D. Howe, J. B. Woodruff, M. A. Miller, G. R. Ax, T. E. Petty, E. A. Sornsin, “Multispectral infrared Stokes imaging polarimeter,” in Polarization: Measurement, Analysis, and Remote Sensing, D. B. Chenault, M. J. Duggin, W. G. Egan, D. H. Goldstein, eds., Proc. SPIE3754, 137–143.

Pérez,

C. Graffigne, F. Heitz, Pérez, F. Prêteux, M. Sigelle, J. Zerubia, “Hierarchical Markov random field models applied to image analysis: a review,” in Neural, Morphological, and Stochastic Methods in Image and Signal Processing, E. R. Dougherty, F. J. Preteux, S. S. Shen, eds., Proc. SPIE2568, 2–17 (1995).
[CrossRef]

Pérez, P.

P. Pérez, A. Chardin, J.-M. Laferté, “Noniterative manipulation of discrete energy-based models for image analysis,” Pattern Recogn. 33, 573–586 (2000).
[CrossRef]

J.-M. Laferté, P. Pérez, F. Heitz, “Discrete Markov image modeling and inference on the quad-tree,” IEEE Trans. Image Process. 9, 390–404 (2000).
[CrossRef]

M. Mignotte, C. Collet, P. Pérez, P. Bouthemy, “Three-class Markovian segmentation of high resolution sonar images,” J. Computer Vision Image Underst. 76, 191–204 (1999).
[CrossRef]

P. Pérez, F. Heitz, “Restriction of a Markov random field on a graph and multiresolution statistical image modeling,” IEEE Trans. Inf. Theory 42, 180–190 (1996).
[CrossRef]

P. Rostaing, P. Pérez, J.-N. Provost, C. Collet, P. Bouthemy, “Multispectral spot images analysis using generalized Gaussian modeling: application to water depth mapping,” Comput. Vis. Image Underst. (to be published).

Petty, T. E.

M. H. Smith, J. D. Howe, J. B. Woodruff, M. A. Miller, G. R. Ax, T. E. Petty, E. A. Sornsin, “Multispectral infrared Stokes imaging polarimeter,” in Polarization: Measurement, Analysis, and Remote Sensing, D. B. Chenault, M. J. Duggin, W. G. Egan, D. H. Goldstein, eds., Proc. SPIE3754, 137–143.

Pieczynski, W.

N. Giordana, W. Pieczynski, “Estimation of generalized multisensor hidden Markov chains and unsupervised image segmentation,” IEEE Trans. Pattern Anal. Mach. Intell. 19, 465–475 (1997).
[CrossRef]

W. Pieczynski, “Statistical image segmentation,” Mach. Graph. Vision 1, 261–268 (1992).

Prêteux, F.

C. Graffigne, F. Heitz, Pérez, F. Prêteux, M. Sigelle, J. Zerubia, “Hierarchical Markov random field models applied to image analysis: a review,” in Neural, Morphological, and Stochastic Methods in Image and Signal Processing, E. R. Dougherty, F. J. Preteux, S. S. Shen, eds., Proc. SPIE2568, 2–17 (1995).
[CrossRef]

Provost, J.-N.

P. Rostaing, P. Pérez, J.-N. Provost, C. Collet, P. Bouthemy, “Multispectral spot images analysis using generalized Gaussian modeling: application to water depth mapping,” Comput. Vis. Image Underst. (to be published).

P. Rostaing, J.-N. Provost, C. Collet, “Unsupervised multispectral image segmentation using generalized Gaussian noise model,” in Proceedings of the International Workshop EMMCVPR’99, Vol. 1654 of Springer Lecture Notes in Computer Science (Springer-Verlag, Berlin, 1999), pp. 141–156.

Rahmann, S.

S. Rahmann, “Polarization images: a geometric interpretation for shape analysis,” in 15th International Conference on Pattern Recognition (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 2000), Vol. 3, pp. 542–546.

Rakovic, M. J.

Réfrégier, P.

F. Goudail, P. Réfrégier, “Statistical algorithms for processing polarimetric images in coherent light,” Trait. Signal 18(5–6), 297–319 (2001).

Roberts, P. J.

Rostaing, P.

P. Rostaing, P. Pérez, J.-N. Provost, C. Collet, P. Bouthemy, “Multispectral spot images analysis using generalized Gaussian modeling: application to water depth mapping,” Comput. Vis. Image Underst. (to be published).

P. Rostaing, J.-N. Provost, C. Collet, “Unsupervised multispectral image segmentation using generalized Gaussian noise model,” in Proceedings of the International Workshop EMMCVPR’99, Vol. 1654 of Springer Lecture Notes in Computer Science (Springer-Verlag, Berlin, 1999), pp. 141–156.

Shapiro, M.

C. A. Bouman, M. Shapiro, “A multiscale random field model for Bayesian image segmentation,” IEEE Trans. Image Process. 3, 162–177 (1994).
[CrossRef] [PubMed]

Sigelle, M.

C. Graffigne, F. Heitz, Pérez, F. Prêteux, M. Sigelle, J. Zerubia, “Hierarchical Markov random field models applied to image analysis: a review,” in Neural, Morphological, and Stochastic Methods in Image and Signal Processing, E. R. Dougherty, F. J. Preteux, S. S. Shen, eds., Proc. SPIE2568, 2–17 (1995).
[CrossRef]

Simon, R.

R. Sridhar, R. Simon, “Normal form for Mueller matrices in polarization optics,” J. Mod. Opt. 41, 1903–1915 (1994).
[CrossRef]

M. S. Kumar, R. Simon, “Characterization of Mueller matrices in polarization optics,” Opt. Commun. 88, 464–470 (1992).
[CrossRef]

Smith, M. H.

P. Y. Gerligand, M. H. Smith, R. A. Chipman, “Polarimetric images of a cone,” Opt. Express 4, 420–430 (1999), http://www.opticsexpress.org .
[CrossRef] [PubMed]

M. H. Smith, J. D. Howe, J. B. Woodruff, M. A. Miller, G. R. Ax, T. E. Petty, E. A. Sornsin, “Multispectral infrared Stokes imaging polarimeter,” in Polarization: Measurement, Analysis, and Remote Sensing, D. B. Chenault, M. J. Duggin, W. G. Egan, D. H. Goldstein, eds., Proc. SPIE3754, 137–143.

M. H. Smith, “Interpreting Mueller matrix images of tissues,” in Laser-Tissue Interaction XII: Photochemical, Photothermal, and Photomechanical, D. D. Duncan, S. L. Jacques, P. C. Johnson, eds., Proc. SPIE4257, 82–89 (2001).
[CrossRef]

Sornsin, E. A.

M. H. Smith, J. D. Howe, J. B. Woodruff, M. A. Miller, G. R. Ax, T. E. Petty, E. A. Sornsin, “Multispectral infrared Stokes imaging polarimeter,” in Polarization: Measurement, Analysis, and Remote Sensing, D. B. Chenault, M. J. Duggin, W. G. Egan, D. H. Goldstein, eds., Proc. SPIE3754, 137–143.

Sridhar, R.

R. Sridhar, R. Simon, “Normal form for Mueller matrices in polarization optics,” J. Mod. Opt. 41, 1903–1915 (1994).
[CrossRef]

Tenney, R.

M. R. Luettgen, W. C. Karl, A. S. Willsky, R. Tenney, “Multiscale representation of Markov random fields,” IEEE Trans. Signal Process. 41, 3377–3395 (1993).
[CrossRef]

van der Mee, C. V. M.

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Figures (11)

Fig. 1
Fig. 1

Schematic of the imaging system used in this study. See text for details.

Fig. 2
Fig. 2

Mueller images of a manufactured object. The image at the upper left corresponds to a conventional intensity image.

Fig. 3
Fig. 3

Mueller images of a plant leaf. The image at the upper left corresponds to a conventional intensity image.

Fig. 4
Fig. 4

Steps in the three algorithms for clustering Mueller images. The Markov-model-based algorithm directly processes 16 Mueller channels (steps α-a-b). The polar decomposition algorithm is mainly a physical processing of polarization data (steps α-c-e). The polarization-contrast enhancement algorithm uses polarization-based information to reduce the data’s dimensionality and to provide a more suitable image for classic image-processing algorithms.

Fig. 5
Fig. 5

Retardance parameter image obtained by polar decomposition. The two shapes are well separated from the background, but the method does not separate the two shapes into two different classes.

Fig. 6
Fig. 6

Result of polarization-contrast enhancement of the manufactured object. This result is to be compared with the m 00 Mueller element image (Fig. 2) and with Fig. 5.

Fig. 7
Fig. 7

Result of polarization-contrast enhancement of the plant leaf. This result is to be compared with the m 00 Mueller element image (Fig. 3). Polar decomposition fails to give a good result for this object.

Fig. 8
Fig. 8

Graph of interlevel dependencies that corresponds to a quad-tree structure. Filled circles, labels; open circles, observations. The segmentation algorithm re-estimates iteratively the parameters of a given hidden in-scale Markov model to produce a new model that has a higher probability of generating the given observation sequence. This re-estimation procedure is continued until no more significant improvement in parameters can be obtained. The two-step computation of posterior marginals propagates available information all over the tree: on one hand, the bottom-up step spreads the influence of data to other levels up to the root; on the other hand, the top-down step backpropagates the information from course to fine levels.

Fig. 9
Fig. 9

Schematic illustration of the Markovian segmentation process used in this study.

Fig. 10
Fig. 10

Label map of the Mueller images of the manufactured target. The Markovian quad-tree segmentation produces a label map with four classes, which are represented by four gray levels (class 0, black; class 1, dark gray; class 2, light gray; class 3, white). The two manufactured objects are effectively well detected in two different (black and white) classes, whereas the background corresponds to light or dark gray pixels. Thus the segmentation process exhibits good properties that allow the two objects to be separated into different classes by use of Mueller element images.

Fig. 11
Fig. 11

Label map of the Mueller images of the plant leaf target. The main innovation in image processing that permits such a result resides in dealing with the whole cube of 16 images of Mueller elements and not directly, as is usual, with the observed luminance. In this example we reach the limit of the Markovian method based on the quad-tree structure. Indeed, block effects appear and disturb the segmentation map. Nevertheless, it is difficult to segment this image in an automated way.

Equations (14)

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

S=s0s1s2s3=ExEx*+EyEy*ExEx*-EyEy*2Ex*Ey2Ex*Ey,
s02s12+s22+s32,
S=Su+Sp=s0-s12+s22+s321/2000+s12+s22+s321/2s1s2s3,
So=MSi,
Iij=121-cos2 2α2k-0.5 sin2 4α2k×sin 2α2kMij1cos2 2α1l0.5 sin2 4α1lsin 2α1l,
Mij=A-1IijP-1,
A=1-cos2 2α21-0.5 sin2 4α21sin 2α211-cos2 2α22-0.5 sin2 4α22sin 2α221-cos2 2α23-0.5 sin2 4α23sin 2α231-cos2 2α24-0.5 sin2 4α24sin 2α24,
P=1111cos2 2α11cos2 2α12cos2 2α13cos2 2α140.5 sin2 4α110.5 sin2 4α120.5 sin2 4α130.5 sin2 4α14sin 2α11sin 2α12sin 2α13sin 2α14.
x=xnn=0R, xn=xs, sSn,
Pxn|xk, k>n=Pxn|xn+1.
Pxn|xn+1=sSn Pxs|xs-,
Py|x=n=0R Pyn|xn=n=0RsSn Pys|xs,
Px, y=Pxrsr Pxs|xs-sS Pys|xs.
s=argmaxωiΔ Pxs=ωi|Y=y.

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