Masses des lois multinomiales négatives. Application au traitement d'images polarimétriques

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Masses des lois multinomiales négatives. Application au traitement d’images polarimétriques

Philippe Bernardoff, Florent Chatelain, Jean-Yves Tourneret

To cite this version:

Philippe Bernardoff, Florent Chatelain, Jean-Yves Tourneret. Masses des lois multinomiales néga-

tives. Application au traitement d’images polarimétriques. 42èmes Journées de Statistique, May

2010, Marseille, France. �inria-00494739�

Masses des lois multinomiales negatives.

Application au traitement d images polarimetriques.

Philippe Bernardo¤, Florent Chatelain et Jean-Yves Tourneret.

P. Bernardo¤ : Université de Pau et des Pays de l’Adour, avenue de l’Uni- versité, 64000 Pau, France. LMAP UMR 5142. E-mail : philippe.bernardo¤

@univ-pau.fr

F. Chatelain : Gipsa-lab, Département Signal et Images, 961 rue de la Houille Blanche, BP 46, 38402 Saint Martin d’Heres, France. E-mail : ‡orent.chatelain@gipsa- lab.inpg.fr

J.-Y. Tourneret : Université de Toulouse, IRIT/ENSEEIHT/T esa, 2 rue Charles Camichel, BP 7122, 31071. Toulouse cedex 7, France

Résumé : Cet article est dérivé d’une nouvelle expression des masses des lois multinomiales négatives multivariées. Cette expression des masses peut être utilisée pour déterminer les estimateurs du maximum de vraisemblance de ses paramètres inconnus. Une application au traitement d’images polarimétriques est étudiée. Plus précisément, les estimateurs du degré de polarisation utilisant la méthode du maximum de vraisemblance avec di¤érentes combinaisons d’images sont comparés.

Mots clés : Lois multinomiales négatives, maximum de vraisemblance,images polarimétriques, degrés de polarisation.

Abstract : This paper derives new closed-form expressions of the masses of multivariate negative multinomiale distributions. These masses can be maximi- zed to determine the maximum likelihood estimator of its unknown parameters.

An application to polarimetric image processing is investigated. More precisely, estimators of the polarization degree using maximum likelihood methods with di¤erent combinations of images are compared.

Keywords : Negative multinomial distributions, maximum likelihood, pola- rimetric images, degree of polarization

1 Introduction

Bar Lev et al. [1] introduced multivariate NMDs whose PGFs are de…ned as the inverse th power of any a¢ ne polynomial as follows. Let us denote [n] = f 1; : : : ; n g and z ^T = Q

t

T z t , where z = (z 1 ; : : : ; z n ) 2 R ⁿ and T [n].

Let P n (z) = P

T [n];T

=

p T z ^T be an a¢ ne polynomial, with respect to the n variables (z 1 ; : : : ; z n ) such as 1 P n (1) 6 = 0: The NMD N M (n; P n ) associated to (n; P n ) is de…ned by its PGF equal to G

) (z) = (1 P n (z)) (1 P n (1)) . These very general multivariate NMDs were recently used for image processing applications in [3].

For a valid NMD, the corresponding expression of the coe¢ cient of z in the Taylor expansion of (1 P _n (z)) is given by the formula (see [2])

c ( ; P _n ) = X

k

K

( )

_k

p ^k

k! ; (1)

where K = f k : P n ! Ng ; and P n is the set of all subsets of [n].

The …rst part of this paper derives a way of computing the masses of mul- tivariate NMDs N M (n; P n ) de…ned above. The second part of the paper is devoted to the application of NMDs to image processing, more speci…cally to polarimetric image processing.

2 Negative Multinomial Distributions

An n-variate NMD is the distribution of a random vector N = (N 1 ; :::; N n ) taking its values in N ⁿ 0 whose PGF is

G

(z) = E Y n k=1

z ^N _k

!

= [P _n (z)] (2)

where E denotes the mathematical expectation, z = (z ₁ ; :::; z _n ); > 0 and P _n (z) is an a¢ ne polynomial of order n

. The a¢ ne polynomial P _n has to sa- tisfy appropriate conditions to ensure that G

(z) is a PGF (see [2]). These conditions include the equality P _n (1; :::; 1) = 1 . As explained in [2], the a¢ ne polynomial P n (z) can be rewritten P n (z) = A n (a 1 z 1 ; :::; a n z n )=A n (a 1 ; :::; a n ) where a 1 ; :::; a n are positive numbers and A n is an a¢ ne polynomial such that A n (0; :::; 0) = 1. The Taylor expansions of [A n (z)] and [P n (z)] in the neigh- borhood of (0; :::; 0) will be denoted as follows

[A n (z)] = X

2Nⁿ0

c ( ; A n )z ; [P n (z)] = X

2Nⁿ0

c ( ; P n )z (3)

where = ( 1 ; :::; n ) and z = Q n

i=1 z _i

. The masses of multivariate NMDs are c ( ; P n ) = c ( ; A n )A n (a 1 ; :::; a n ) Q n

i=1 a _i

:

3 Masses of Negative Multinomial Distributions

Before computing the c ( ; A n ), we need the following result

Theorem 1 Denote as P _n the set of non empty subsets of [n] = f 1; :::; n g . Any a¢ ne polynomial A n such that A n (0) = 1 denoted as A n (z) = 1 P

T

a T z ^T : Moreover

A n (z) = 2 4 Y

i

[n]

(1 a i z i ) 3

5 1 Q n

z 1

1 a 1 z 1

; : : : ; z n

1 a n z n

(4)

where j T j is the cardinal of the set T; and Q n is the polynomial de…ned by Q _n (z) = P

T

;

T

2 d ⁿ _T z ^T and d ⁿ _T is related to the 2

^T

1 variables a _S ; S 2 P _T as follows d ⁿ _T = P

T

T

;

T

>1 a T a ^[n]

^T + ( j T j 1) Q

i

T a i :

1

A polynomial P

(z) with respect to z = (z

; : : : ; z

) is a¢ ne if the one variable polynomial

z

7! P

(z) can be written A

z

+ B

(for any j = 1; : : : ; d), where A

and B

are polynomials with respect to the z

’s with i 6 = j.

Theorem 2 Let A _n (z) = 1 P

T

a _T z ^T ; a = (a ₁ ; : : : ; a _n ) and Q _n the a¢ ne polynomial de…ned in Theorem 1. For any and in N ⁿ ; denote as c ( ; A _n ) the coe¢ cient of z in the Taylor expansion of [A n (z)] and c ( ; 1 Q n ) the coe¢ cient of z in the Taylor expansion [1 Q n (z)] . The following relation can be obtained

c ( ; A n ) = X

+ =

c ( ; 1 Q n ) ( 1 + ) a

! (5)

= X

0

; i=1;:::;n

c ( ; 1 Q _n ) Y n i=1

( + _{i i} )

a _i

i ! (6)

= X

0

; i=1;:::;n

c ( ; 1 Q _n ) Y n i=1

( + _i )

i

a _i

( i i )! (7)

3.1 Bivariate NMDs

Theorem 3 The coe¢ cient of z in the Taylor expansion of [A ₂ (z)] ; can be computed as follows

c ( ; P 2 ) = ( ) _max(

_;

₎

min(

;

)

X

`=0

( + `) _min(

_;

_{) `}

( 1 `)! ( 2 `)!`! a ₁

^` a ₂

^{`</sup> b <sup>`} _1;2 : (8)

3.2 Trivariate NMDs

Theorem 4 The coe¢ cient of z in the Taylor expansion of [A 3 (z)] can be expressed as follows

c ( ; P ₃ ) = X

1

=0

X

=0

X

=0

b

X

c

v=

( ) _v b ^v _2;3

⁺

b ^v _1;3

⁺

b ^v _1;2

⁺

Q 3

i=1 (v i + i )!

b

_1;2;3

^2v ( j j 2v)!

( + _{1 1} )

1 !

( + _{2 2} )

2 !

( + _{3 3} )

3 ! a ₁

a ₂

a ₃

: (9)

4 Estimating the polarization degree of low-‡ux polarimetric images using maximum likelihood methods

4.1 Low-‡ux polarimetric images

The state of polarization of the light can be described by the random beha- vior of a complex vector A = (A X ; A Y ), called the Jones vector, whose cova- riance matrix, called the polarization matrix, is

= E (A X A _X ) E (A X A _Y )

E (A _Y A _X ) E (A _Y A _Y ) = a 1 a 3 + ia 4

a ₃ ia ₄ a ₂

where denotes the complex conjugate. The covariance matrix is a non ne-

gative hermitic matrix whose diagonal terms are the intensity components in

the X and Y directions. The cross terms of are the correlation between the Jones components. If we assume a fully developed speckle, the Jones vector A is distributed according to a complex Gaussian distribution with probability den- sity function (pdf) (see [5]) p(A) = 1=( ² j j ) exp A

¹ A where j j is the determinant of the matrix and

denotes the conjugate transpose operator.

As a consequence, the statistical properties of A are fully characterized by the covariance matrix . The di¤erent components of can be classically estimated by using four intensity images that are related to the components of the Jones vector as follows (see [3], for more details)

I 1 = j A X j ² ; I 2 = j A Y j ² ; I 3 = 1

2 j A X j ² + 1

2 j A Y j ² + Re (A X A _X ) ; I 4 = 1

2 j A X j ² + 1

2 j A Y j ² + Im (A X A _Y ) :

The state of polarization of the light is classically characterized by the square DoP de…ned by ([5])

P ² = 1 4 j j

[trace( )] ² = 1 4 a 1 a 2 (a ² ₃ + a ² ₄ )

(a 1 + a 2 ) ² (10) where trace( ) is the trace of the matrix . The light is totally depolarized for P = 0, totally polarized for P = 1 and partially polarized when P 2 ]0; 1[.

Di¤erent estimation methods of P ² using several combinations of intensity images were studied in [3]. Since only one realization of the random vector I = (I ₁ ; :::; I ₄ ) ^T was available for a given pixel of a polarimetric image, the image was supposed to be locally stationary and ergodic. These assumptions were used to derive square DoP estimators using several neighbor pixels belon- ging to a so-called estimation window.

This section considers practical applications where the intensity level of the re‡ected light is very low (low-‡ux assumption), which leads to an additional source of ‡uctuations on the detected signal. Under the low-‡ux assumption, the quantum nature of the light leads to a Poisson-distributed noise which can become very important relatively to the mean value of the signal at low photon level. As a consequence, the observed pixels of the low-‡ux polarimetric image are discrete random variables contained in the vector N = (N 1 ; :::; N 4 ) such that the conditional distributions of the random variables N _l j I _l , for l = 1; :::; 4 are in- dependent and distributed according to Poisson distributions with means I _l , for l = 1; :::; 4. The resulting joint distribution of N is a multivariate mixed Poisson distribution (see [4]) P (N = k) = R R

(

)

Q 4 l=1

I

k

! exp ( I _l ) f (I) dI, where k = (k 1 ; :::; k 4 ), k i 2 N and f (I) is the joint pdf of the intensity vector. This section studies estimators of the square DoP P ² de…ned in (10) based on several vectors N ¹ ; : : : ; N ⁿ belonging to the estimation window.

The joint distribution of the intensity vector I is known to be a multivariate gamma distribution whose Laplace transform is (see [3]) E h

exp P 4

k=1 z _k I _k i

= 1=P 4 (z) where the a¢ ne polynomial P 4 is P 4 (z) = 1 + z + k a [2z 1 z 2 + z 3 z 4 + (z 1 + z 2 )(z 3 + z 4 )] with z = (z 1 ; :::; z 4 ) and k a = ¹ ₂ a 1 a 2 a ² ₃ a ² ₄ ; = (a ₁ ; a ₂ ; a ₃ + (a ₁ + a ₂ ) =2; a ₄ + (a ₁ + a ₂ ) =2) ^T : As a consequence, the distribu- tion of N is an NMD whose PGF can be written

G

(z) = [P 4 (z 1 1; z 2 1; z 3 1; z 4 1)] ¹ : (11)

The results of Section 3 allow us to compute the masses of N that will be useful for studying the maximum likelihood estimator (MLE) of the square DoP.

4.2 MLE using three polarimetric images

The PGF of f N = (N 1 ; N 2 ; N 3 ) can be computed from (11) by setting z 4 = 1.

The following result can be obtained

G

(z) = [P ₃ (z)] ¹ (12)

with P ₃ (z) = P ₃ (0)+z ₁ ( ₁ 3k _a )+z ₂ ( ₂ 3k _a )+z ₃ ( ₃ 2k _a )+k _a (2z ₁ z ₂ +z ₁ z ₃ + z 2 z 3 ); z = (z 1 ; z 2 ; z 3 ) and P 3 (0) = 1 P 3

i=1 i + 4k a . The results of Section 3.2 can then be used to express the masses of N f as a function of = (a 1 ; a 2 ; a 3 ; a ² ₄ ) ^T . The ML estimator of based on several vectors N f ^k belonging to the estima- tion window (where k = 1; :::; K and K is the number of pixels of the observa- tion window) is obtained by maximizing the log-likelihood l ₃ N f ¹ ; :::; f N ^K j = P K

k=1 log h

P f N ^k i

with respect to . The practical determination of the ML estimator of is achieved by using a Newton-Raphson procedure. The ML es- timators of the vector elements, denoted as e = e a 1 ; e a 2 ; e a 3 ; a e ² ₄

T

, are then plugged into (10) to provide the ML estimator of the square DoP based on three polarimetric images

P e ² = 1 4 h

e a ₁ e a ₂ ( e a ² ₃ + a e ² ₄ ) i

( e a ₁ + e a ₂ ) ² : (13)

4.3 MLE using two polarimetric images

The PGF of N = (N ₁ ; N ₂ ) can be computed from (12) by setting z ₃ = 1.

The following result can be obtained

G

(z) = [P ₂ (z)] ¹ (14)

with P 2 (z) = P 2 (0) + z 1 ( 1 2k a ) + z 2 ( 2 2k a ) + 2k a z 1 z 2 ; z = (z 1 ; z 2 ) and P 2 (0) = 1 P 2

i=1 i +2k a . The results of Section 3.1 can then be used to express the masses of N as a function of = (a ₁ ; a ₂ ; k _a ) ^T .

The ML estimator of based on several vectors N ^k belonging to the estima- tion window is obtained by maximizing the log-likelihood l 2 N ¹ ; :::; N ^P j = P K

k=1 log h

P N ^k i

with respect to . The practical determination of the ML estimator of is achieved by using a Newton-Raphson procedure. The ML esti- mators of the vector elements, denoted as = a ₁ ; a ₂ ; k a

T , are then plugged into (10) to provide the ML estimator of the square DoP based on two polari- metric images

P ² = 1 8k _a

(a ₁ + a ₂ ) ² : (15)

5 Simulations results

Simulations results showing the estimation performance and illustrating the application to polarimetric imagery will be presented during the conference.

Acknowledgements

The authors would like to thank Gérard Letac for fruitful discussions regar- ding multivariate gamma distributions and negative multinomial distributions.

Références

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Journal of Theoretical Probability, 7(4) :883–929, October 1994.

[2] Philippe Bernardo¤. Which negative multinomial distributions are in…nitely divisible ? Bernoulli, 9(5) :877–893, 2003.

[3] F. Chatelain, J.-Y. Tourneret, M. Roche, and M. Alouini. Estimating the polarization degree of polarimetric images in coherent illumination using maximum likelihood methods. JOSA A, 26(6) :1348–1359, June 2009.

[4] A. Ferrari, G. Letac, and J.-Y. Tourneret. Multivariate mixed Poisson dis- tributions. In F. Hlawatsch, G. Matz, M. Rupp, and B. Wistawel, editors, EUSIPCO-04, pages 1067–1070, Vienna, Austria, September 2004.

[5] J. Goodman. Statistical Optics. Wiley, New York, 1985.