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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(

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

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

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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

2020

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.