A JOINT TEST STATISTIC CONSIDERING COMPLEX WISHART DISTRIBUTION 
CHARACTERIZATION OF TEMPORAL POLARIMETRIC DATA 
Esra Erten, Andreas Reigber, Rafael Zandonà Schneider and Olaf Hellwich 
Computer Vision and Remote Sensing, Technical University of Berlin 
D-10587, Berlin, Germany. 
Microwaves and Radar Institute, German Aerospace Centre (DLR) 
D-82234 Oberpfaffenhofen, Germany. 
esra.erten@dlr.de 
Commission VII 2/1 Information Extraction from SAR data 
KEY WORDS: polarimetrie SAR, temporal analysis, tracking, change detection, complex wishart 
ABSTRACT: 
Polarimetrie data of distributed scatterers can be fully characterized by the (3 x 3) Hermitian positive definite matrix which follows a 
complex Wishart distribution under Gaussian assumption. A second observation in time will also follow Wishart distribution. Then, 
these observations are correlated or uncorrelated process over time related to the monitored objects. To not to make any assumption 
concerning their independence, the (6 x 6) matrix which is also modeled as a complex Wishart distribution is used in this study to 
characterize the behavior of the temporal polarimetrie data. According to the complex density function of (6 x 6) matrix, the joint 
statistics of two polarimetrie observation is extracted. The results obtained in terms of the joint and the marginal distributions of Wishart 
process are based on the explicit closed-form expressions that can be used in pdf (probability density function) based statistical analysis. 
Especially, these statistical analysis can be a key parameter in target detection, change detection and SAR sequence tracking problem. 
As demonstrated the bias of the joint distribution can decrease with noise free signal and with increasing the canonical correlation 
parameter, number of looks and number of acquired SAR images. The results of this work are analyzed by means of simulated data. 
1 INTRODUCTION 
tribution of decomposed scattering mechanism as follows 
In this paper, the joint and the marginal statistics of a temporal 
polarimetrie data (correlated complex Wishart process over time) 
is studied. There appears to be very little published work in the 
context of polarimetrie data, although (Martinez et al., 2005) con 
tains similar statistical analysis, focusing on only one polarimet 
rie data rather than multi-temporal data set. 
The polarimetric SAR measures the amplitude and phase of scat 
tered signals in combination of the linear receive and transmit 
polarizations. This signals from the complex scattering mecha 
nism are related to the incident and scattered Jones vectors. Using 
a straightforward lexicographic ordering of the scattering matrix 
elements, a complex target vector k — [Shh Shv is ob 
tained for backscattering case 1 , and it can be modeled as a mul 
tivariate complex Gaussian pdf A/’ c (0, £) with £ = £{kk t }. 
The inherent speckle in SAR data can be decreased by indepen 
dent (uncorrelated pixels) averaging techniques with the cost of 
decreasing resolution. In this so called multilook case, (k *) 
follows a complex Wishart pdf W c (n, £) (Laurent et al., 2001) 
with the degrees of freedom n and covariance matrix £ where 
f indicates the conjugate transpose operator. The components of 
covariance matrix contains all scattering matrix elements as 
(£> = 
(ShhS hh ) 
(ShvS^h) 
.(SvvS^b) 
(ShhSl) 
(ShvSI) 
(SvvSfo) 
(1) 
and the decomposition theorem of covariance matrix allows to 
create a set of orthonormal (independent) scattering mechanisms, 
whereas the corresponding eigenvalue express the individual con- 
1 without considering the constants for the power conservation when 
changing from the 4 dimensional to the 3 dimension k polarimetric ac 
quisition vector 
U = [ei, e2,63], unit eigenvectors 
A = 5Z i=1 Ai(ei.ej). 
(2) 
Considering the potential of target decomposition (TD) in polari 
metric parameter estimation, the joint and marginal distribution 
of eigenvalues of target vectors are discussed in coming sections. 
2 CHARACTERIZATION OF TEMPORAL DATA SET 
2.1 Derivation of the joint density of two matrices 
To be precise, with the same notations in (Laurent et al., 2001), 
let w = [ki k2] T be a complex target vector distributed as 
multivariate complex Gaussian that consist of two target vectors 
ki and k2 obtained from temporal images at time t\ and £2- The 
joint statistics A = £ J2j=i w j w j has a complex Wishart distri 
bution with n degrees of freedom. Here, q represents the number 
of elements in one of the target vector k, and the vector w has 
the dimension of p = 2q. The n look covariance matrix A sum 
maries whole (joint and marginal) information from both images. 
If A is partioned as A = 
, the conditioned on An, 
An A12 
A21 A22 
the joint density of element A22 follows the complex Wishart 
distribution An. 2 = An - Ai 2 A2 2 1 A2i ~ p(An|A22) = 
Wq (n — q, £11.2) (Laurent et al., 2001), and it is independent 
from A12 and A22- Then, using the well known rule that the 
conditional distribution of correlation matrix A12 given A22 is a 
complex normal distribution p( A121A22) ~A/’ qX q (£ 12 £ f 2 X A22, 
£11.2 ® A22) where ® indicates Kroneker products and the theo 
rem 10.3.2 in (Muirhead, 1982), the conditional distribution of R 2 = 
A^Afj 1 A^ 1 A21 on A22 (p(R 2 1A22)) is a noncentral Wishart 
distribution. Sincep (An.2, A22, R 2 ) = p(An.2)p (R 2 |A22) p(A22),