1. 1 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

2. 2 K-MEANS CLASSIFIER Conventional clustering algorithm:  Initialisation: Assign pixels to classes.  Centers computation: Compute the centers of each class as follows:  Reassignment: Reassign each pixel to the class whose center minimizes a certain distance.

3. OUTLINE 1.Non-Gaussian clutter model: the SIRV model 2.Contribution of the geometry of information 3.Results on real data 4.Conclusions and perspectives

4. OUTLINE 1.Non-Gaussian clutter model : the SIRV model 2.Contribution of the geometry of information 3.Results on real data 4.Conclusions and perspectives

5. 5 CONVENTIONAL COVARIANCE MATRIX ESTIMATE  With low resolution, clutter is modeled as a Gaussian process.  Estimation of the covariance matrix of a pixel, characterized by its target vector k, thanks to N secondary data: k 1, …, k N.  Maximum Likelihood estimate of the covariance matrix, the Sample Covariance Matrix (SCM):

6. 66 SCM IN HIGH RESOLUTION Gamma classificationWishart classification with SCM Results are very close from each other : influence of polarimetric information ?

7. 77 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 pdf is not specified : large class of stochastic processes can be described.  Texture : local spatial variation of power.  Speckle : polarimetric information.  Validated on real data measurement campaigns.

8. 8 COVARIANCE MATRIX ESTIMATE : THE SIRV MODEL 88 ML ESTIMATE UNDER SIRV ASSUMPTION 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 asymptotically Wishart-distributed.

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

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

11. 11 OUTLINE 1.Non-Gaussian clutter model: the SIRV model 2.Contribution of the geometry of information 3.Results on real data 4.Conclusions and perspectives

12. 12 Euclidian Mean: CONVENTIONAL MEAN OF COVARIANCE MATRICES The mean in the Euclidean sense of n given positive-definite Hermitian matrices M 1,..,M n in P(m) is defined as: Barycenter:

13. 13 Riemannian Mean: 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 M 1,..,M n in P(m) is defined as: Geodesic: Riemannian distance:

14. 14 OUTLINE 1.Non-Gaussian clutter model : the SIRV model 2.Contribution of the geometry of information 3.Results on real data 4.Conclusions and perspectives

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

16. 16 CLASSES IN THE H-α PLANE

17. 17 PARACOU, FRENCH GUIANA  Acquired with the ONERA SETHI system  UHF band  1.25m resolution

18. 18 CLASSIFICATION RESULTS Classification with Wishart distance, Arithmetical mean Classification with Wishart distance, Geometrical mean Classification with geometric distance, Geometrical mean

19. 19 OUTLINE 1.Non-Gaussian clutter model : the SIRV model 2.Contribution of the geometry of information 3.Results on real data 4.Conclusions and perspectives

20. 20 CONCLUSIONS  Further investigation of the distance is required.  Interpretation is difficult because no literature.  Span can give information for homogeneous areas.  Further investigation of the distance is required.  Interpretation is difficult because no literature.  Span can give information for homogeneous areas.  Necessity of a non-Gaussian model for HR SAR images.  Geometric definition of the class centers in line with the structure of the covariance matrices space.  Necessity of a non-Gaussian model for HR SAR images.  Geometric definition of the class centers in line with the structure of the covariance matrices space.