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P. Formont 1,2 , J.-P. Ovarlez 1,2 , F. Pascal 2 , G. Vasile 3 , L. Ferro-Famil 4

ON THE EXTENSION OF THE PRODUCT MODEL IN POLSAR PROCESSING FOR UNSUPERVISED CLASSIFICATION USING INFORMATION GEOMETRY OF COVARIANCE MATRICES. P. Formont 1,2 , J.-P. Ovarlez 1,2 , F. Pascal 2 , G. Vasile 3 , L. Ferro-Famil 4 1 ONERA, 2 SONDRA, 3 GIPSA-lab, 4 IETR. K-MEANS CLASSIFIER.

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P. Formont 1,2 , J.-P. Ovarlez 1,2 , F. Pascal 2 , G. Vasile 3 , L. Ferro-Famil 4

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  1. ON THE EXTENSION OF THE PRODUCT MODEL IN POLSAR PROCESSING FOR UNSUPERVISED CLASSIFICATION USING INFORMATION GEOMETRY OF COVARIANCE MATRICES P. Formont1,2, J.-P. Ovarlez1,2, F. Pascal2, G. Vasile3, L. Ferro-Famil4 1 ONERA, 2 SONDRA, 3 GIPSA-lab, 4 IETR

  2. K-MEANS CLASSIFIER • Conventionalclusteringalgorithm: • Initialisation: Assign pixels to classes. • Centers computation: Compute the centers of each class as follows: • Reassignment: Reassigneach pixel to the class whose center minimizes a certain distance.

  3. OUTLINE Non-Gaussian clutter model: the SIRV model Contribution of the geometry of information Results on real data Conclusions and perspectives

  4. OUTLINE Non-Gaussianclutter model : the SIRV model Contribution of the geometry of information Results on real data Conclusions and perspectives

  5. CONVENTIONAL COVARIANCE MATRIX ESTIMATE • Withlowresolution, clutterismodeled as a Gaussianprocess. • Estimation of the covariance matrix of a pixel, characterized by itstargetvectork, thanks to N secondary data: k1, …, kN. • Maximum Likelihoodestimate of the covariance matrix, the Sample Covariance Matrix (SCM):

  6. SCM IN HIGH RESOLUTION Gamma classification Wishart classification with SCM Results are very close fromeachother : influence of polarimetric information ? 6

  7. THE SIRV MODEL Non-Gaussian SIRV (Spherically Invariant Random Vector) representation of the scattering vector : • where is a random positive variable (texture) and (speckle). • The texture pdfis not specified : large class of stochasticprocessescanbedescribed. • Texture : local spatial variation of power. • Speckle : polarimetric information. • Validated on real data measurementcampaigns. 7

  8. COVARIANCE MATRIX ESTIMATE : THE SIRV MODEL ML ESTIMATE UNDER SIRV ASSUMPTION COVARIANCE MATRIX ESTIMATE : THE SIRV MODEL • Under SIRV assumption, the SCM is not a good estimator of M. • ML estimate of the covariance matrix: • Existence and unicity. • Convergence whatever the initialisation. • Unbiased, consistent and asymptoticallyWishart-distributed. 8 8

  9. DISTANCE BETWEEN COVARIANCE MATRICES UNDER SIRV ASSUMPTION • Gaussian Process ↔ Generalized LRT ↔ Wishart distance between the two SCM covariance matrices • Non Gaussian Process ↔ Generalized LRT ↔ SIRV distance between the two FP covariance matrices

  10. COVARIANCE MATRIX ESTIMATE : THE SIRV MODEL RESULTS ON REAL DATA COVARIANCE MATRIX ESTIMATE : THE SIRV MODEL Color composition of the region of Brétigny, France Wishart classification with SCM Wishart classification with FPE 10 10 10

  11. OUTLINE Non-Gaussianclutter model: the SIRV model Contribution of the geometry of information Results on real data Conclusions and perspectives

  12. CONVENTIONAL MEAN OF COVARIANCE MATRICES The mean in the Euclidean sense of n given positive-definite Hermitian matrices M1,..,Mn in P(m) is defined as: Barycenter: Euclidian Mean: 12

  13. A DIFFERENTIAL GEOMETRIC APPROACH TO THE GEOMETRIC MEAN OF HERMITIAN DEFINITE POSITIVE MATRICES The mean in the Riemannian sense of n given positive-definite Hermitian matrices M1,..,Mn in P(m) is defined as: Geodesic: Riemannian Mean: Riemannian distance:

  14. OUTLINE Non-Gaussianclutter model : the SIRV model Contribution of the geometry of information Results on real data Conclusions and perspectives

  15. CLASSIFICATION RESULTS Wishart classification with SCM, Arithmeticalmean SIRV classification with FPE, Arithmeticalmean SIRV classification with FPE, Geometrical mean

  16. CLASSES IN THE H-α PLANE

  17. PARACOU, FRENCH GUIANA • Acquiredwith the ONERA SETHI system • UHF band • 1.25m resolution 17

  18. CLASSIFICATION RESULTS Classification withWishart distance, Arithmeticalmean Classification withWishart distance, Geometricalmean Classification withgeometric distance, Geometrical mean 18

  19. OUTLINE Non-Gaussianclutter model : the SIRV model Contribution of the geometry of information Results on real data Conclusions and perspectives

  20. CONCLUSIONS • Necessity of a non-Gaussian model for HR SAR images. • Geometricdefinition of the class centers in line with the structure of the covariance matrices space. • Further investigation of the distance isrequired. • Interpretationisdifficultbecause no literature. • Spancangive information for homogeneous areas. 20

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