The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part Bl. Beijing 2008 
132 
Application I: The determinant of the covariance matrix is the 
generalized variance of polarimetric data, and the ratio of two 
determinants is an important parameter in applications as in edge 
detection (Skriver et al., 2001), change detection (Conradsen2003 
et al., 2003) and SAR image tracking (Erten et al., 2008). An un 
biased estimator of the ratio of two covariance matrices determi 
nants j^j is given by . The expectation of is derived 
from (3) as a function of the canonical correlation coefficients 
r = [p\ p2 p 3 ] T . For that, a same procedure as in (Lliopoulos, 
2006) is applied into (3), and the unbiased estimate of jfj^j for 
n > 2q results 
lAn[(n-p)(n-g) 1 
|A 22 |(n-ç- 1) 
^2(n — p — 1) + 
+ 
n — 2q J 
(18) 
To obtain (18) from (3), integrations over An and A 22 are nec 
essary. These integrations are only valid if the variance of obser 
vations are finite, or in other words, the condition of | An | < oo 
and IA 22 1 < oo which are always satisfied in the polarimetric 
case. Figure 2 presents the evaluation of the unbiased estimate of 
related to the number of looks n and the canonical corre 
lation coefficients r. It is clear that the generalized variance ratio 
is asymptotically unbiased for a large number of looks and the 
o 
-P 
Figure 2: The expected value of the polarimetric variance ratio 
estimator with r = [p\ p 2 p$\ T 
Application II: As indicated in (Martinez et al., 2005), the proba 
bility of the maximum eigenvalue related to the dominant scatter 
ing is a important parameter for target detection and its analysis. 
The probability of the detection that there is just one dominant 
scattering mechanism is a function of the detection threshold T, 
that can be obtained from (15) as 
F\max{T) =P(Amax < T) = / px max (x)dx (19) 
Jo 
where Fx (x) indicates the cumulative density function. In addi 
tion, having the closed form of the eigenvalue pdf (15), the prob 
ability of false alarm can be also computed, as well as receiver 
operating characteristic (ROC) curves, allowing a complete de 
tection problem analysis. 
Application III: Another application of (3) has been detailed ex 
plained in (Erten et al., 2008) with the aim of multidimensional 
SAR tracking. The joint distribution of polarimetric covariance 
matrices over time has been used to perform maximum likelihood 
tracking into amplitude data without making any assumption of 
their independence. 
4 CONCLUSIONS AND FUTURE WORK 
The statistical description of two (possible) correlated Wishart 
distributions has been presented. Closed forms for the general 
distribution have been derived, as well as for the joint distribu 
tions of the eigenvalues of the two Wishart matrices and for the 
joint distribution of their maximum eigenvalue. This analysis can 
be applied to a wide field of aplications, whenever the applica 
tion in question follows the statistical assumptions. Examples of 
applications have been given in the paper, as the assessment of 
different aspects in polrimetric statistical analysis over time. It 
has been showed that the performance of analysis may increase 
with high canonical correlation coefficients, the number of look 
and number of polarimetric channel or may decrease due to the 
presence of speckle that effects the calculation of true values of 
the parameters. 
REFERENCES 
Conradsen, K., Nielsen, A. A., Schou, J. and Skriver, H.,2003. A 
test statistic in the complex Wishart distribution and its applica 
tion to change detection in polarimetric SAR data IEEE Trans, 
on Geoscience and Remote Sensing, 41, 1, pp. 4—19. 
Erten, E., Reigber, A., Hellwich, O. and Prats, P., 2007. Indepen 
dency preserving dependent maximum likelihood texture track 
ing model EUSAR 2008, in procedings. 
Gross, K.I, Richards, D.S.P., 1989. Total positivity, spherical se 
ries, and hypergeoometric functions of matrix argument Journal 
of Approx. Theory, 59, pp. 224-246. 
James, A.T., 1964. Distribution of matrix variates and latent roots 
derived from normal samples The Annals of Mathematical Statis 
tics, 35, 2, pp. 475-501. 
Kuo, P. H., Smith, P.J. and Garth, L.M., 2007. Joint den 
sity for eigenvalues of two correlated complex wishart matri- 
cesxharacterization of MIMO systems IEEE Trans, on Wireless 
Communications, 6, 11, pp. 3902-3906. 
Famil, L. F., Pottier, E. and Lee, J. S., 2001. Unsupervised clas 
sification of multifrequency and fully polarimetric SAR images 
based on the H/A/Alpha-Wishart Classifier IEEE Trans, on Geo 
science and Remote Sensing, 39, 11, pp. 2332-2342. 
Lliopoulos, G., 2006. UMVU estimation of the ratio of pow 
ers of normal generalized variances under correlation Journal of 
Multivariate Analysis, available online. 
Martinez, C. L., Pottier, E. and Cloude S. R., 2005. Statistical 
assessment of eigenvector-based target decomposition theorems 
in radar polarimetry IEEE Trans, on Geoscience and Remote 
Sensing, 43, pp. 2058-2074. 
McKay, M.R., Grant, A.J. and Collings, I.B., 2007. Performance 
analysis of MIMO-MRC in Double-Correlated Rayleigh Envi 
ronments IEEE Trans, on Communications, 55, 3, pp. 497-507. 
Muirhead, R. J., 1982. Aspects of Multivariate Statistical Theory 
New York: Wiley. 
Skriver, H., Schou, J., Nielsen, A. A. and Conradsen, K., 2001. 
Polarimetric edge detector based on the complex Wishartdistri- 
bution Geoscience and Remote Sensing Symposium, 2001, 7, 
pp. 3149-3151. 
Smith, P.J. and Garth, L.M., 2007. Distribution and characteris 
tic functions for correlated complex Wishart matrices Journal of 
Multivariate Analysis, 98, pp. 661-667.