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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

N

1

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 N

m

N

m

11

11 ˆˆ

ˆxMx

xx

kMk

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

1010

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

1212

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.

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.