p. formont 1,2 , j.-p. ovarlez 1,2 , f. pascal 2 , g. vasile 3 , l. ferro-famil 4
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
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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
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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.
kixik x
N1
ijxxx jikiki ),d(),d(
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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
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OUTLINE
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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: k1, …, kN.
Maximum Likelihood estimate of the covariance matrix, the Sample Covariance Matrix (SCM):
N
i
HiiSCM N 1
1ˆ kkΤ
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SCM IN HIGH RESOLUTION
Gamma classification Wishart classification with SCM
Results are very close from each other : influence of polarimetric information ?
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THE SIRV MODEL
Non-Gaussian SIRV (Spherically Invariant Random Vector) representation of the scattering vector :
xk
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.
),(~ M0x CN
k
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COVARIANCE MATRIX ESTIMATE : THE SIRV MODELCOVARIANCE MATRIX ESTIMATE : THE SIRV MODEL
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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.
N
i iFPHi
Hii
N
i iFPHi
Hii
FP Nm
Nm
11
11 ˆˆ
ˆxMx
xxkMk
kkM
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DISTANCE BETWEEN COVARIANCE MATRICES UNDER SIRV ASSUMPTION
• Non Gaussian Process ↔ Generalized LRT ↔ SIRV distance SIRV distance between the two FP between the two FP covariance matricescovariance matrices
• Gaussian Process ↔ Generalized LRT ↔ Wishart distance between the two SCM Wishart distance between the two SCM covariance matricescovariance matrices
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COVARIANCE MATRIX ESTIMATE : THE SIRV MODELCOVARIANCE MATRIX ESTIMATE : THE SIRV MODEL
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RESULTS ON REAL DATA
Color composition of the region of Brétigny, France
Wishart classification with SCM Wishart classification with FPE
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OUTLINE
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Euclidian Mean:
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:
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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 M1,..,Mn in P(m) is defined as:
Geodesic:
Riemannian distance: )log(,dR ABBA 1
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OUTLINE
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CLASSIFICATION RESULTS
Wishart classification with SCM, Arithmetical mean
SIRV classification with FPE, Arithmetical mean
SIRV classification with FPE, Geometrical mean
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CLASSES IN THE H-α PLANE
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PARACOU, FRENCH GUIANA
Acquired with the ONERA SETHI system
UHF band
1.25m resolution
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CLASSIFICATION RESULTS
Classification with Wishart distance, Arithmetical mean
Classification with Wishart distance, Geometrical mean
Classification with geometric distance, Geometrical mean
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OUTLINE
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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.