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),
