Formont.ppt
- 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. 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 Non-Gaussian clutter model: the SIRV model Contribution of the geometry of information Results on real data Conclusions and perspectives
- 4. OUTLINE
- 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. SCM IN HIGH RESOLUTION Gamma classification Wishart classification with SCM Results are very close from each other : influence of polarimetric information ?
- 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 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. COVARIANCE MATRIX ESTIMATE : THE SIRV MODEL COVARIANCE MATRIX ESTIMATE : THE SIRV MODEL 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. DISTANCE BETWEEN COVARIANCE MATRICES UNDER SIRV ASSUMPTION Non Gaussian Process ↔ Generalized LRT ↔ SIRV distance between the two FP covariance matrices Gaussian Process ↔ Generalized LRT ↔ Wishart distance between the two SCM covariance matrices
- 10. COVARIANCE MATRIX ESTIMATE : THE SIRV MODEL COVARIANCE MATRIX ESTIMATE : THE SIRV MODEL RESULTS ON REAL DATA Color composition of the region of Brétigny, France Wishart classification with SCM Wishart classification with FPE
- 11. OUTLINE
- 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. 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. OUTLINE
- 15. CLASSIFICATION RESULTS Wishart classification with SCM, Arithmetical mean SIRV classification with FPE, Arithmetical mean SIRV classification with FPE, Geometrical mean
- 16. CLASSES IN THE H- α PLANE
- 17. PARACOU, FRENCH GUIANA Acquired with the ONERA SETHI system UHF band 1.25m resolution
- 18. CLASSIFICATION RESULTS Classification with Wishart distance, Arithmetical mean Classification with Wishart distance, Geometrical mean Classification with geometric distance, Geometrical mean
- 19. OUTLINE
- 20. CONCLUSIONS 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.