The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part Bl. Beijing 2008 
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Consequently, denoting the eigenvalues of An = (kik|) n and 
A22 = ^kJ^n by Aj, A2, A3 and A", X'i, A3, respectively, us 
ing the equation 92 in (James, A.T., 1964), the joint density of 
eigenvalues can be obtained as in the following 
p( Vi, Ai, A',', Aï, Aï) = 0F1 In, 
X A(An)A(A 22 ) x TT 9 ( y l ~ Q 
A ACE^A^^r,,^ n?(l-P?) n 1A * V 7 ! 7 ^ ) 
xex P (-EL ,Ä + Ä) (,1) 
where A m (.) indicate Vandermonde determinants of matrices 
Q 
Aq(S) = ~ a i) 
i<j 
and I n {x) is the modified Bessel function of the first kind of order 
n. 
2.3 Derivation of the joint density of the maximum eigen 
values from temporal images 
In general, as in (Conradsen2003 et al., 2003), the temporal anal 
ysis based on the likelihood ratio test is performed by testing the 
null hypothesis that all the latent roots of An and A22 are equal. 
If this hypothesis is accepted it can be concluded that all the scat 
tering mechanism have same variance over time and hence con 
tribute equally to the total change. It means also that there is no 
need to perform TD with the aim of dimension reductic 
ever, in practice, it is reasonable to consider the null h; 
that deals with the comparison between individual eige 
related to some specific scattering mechanism from diff 
ages) rather than all eigenvalues at once. 
Therefore, to analyze the variation of maximum eigenval 
is related to the dominant scattering mechanism after sc 
the joint pdf of p(Ai, A'/) is required. To compute p 
(A^ A3) and (A2, A3) must be integrated out from (11)' 
p 00 p OO P OO POO 
P(A / i,Ax)= / / / / p(A'i, A2, A3, A'/, A2, 
Jo Jo Jo Jo 
dA^dAs'dAgd 
In addition, the probability density function of the rat 
joint density to the marginal density 
p(Aj, A2, A3, Ai, A2, A3) 
p(Ai,A") 
can be used to analyze the contribution of specific scatter 
anism in respect to the whole temporal scattering mech; 
The whole procedure explained above can be performed even for 
systems with larger dimensions. However, for large multidimen 
sional systems, the large number of integration process related to 
the number of eigenvalues may become complicated. 
2.4 Derivation of the marginal density of the maximum eigerr 
values 
The last statistical analysis is performed with the aim of char 
acterizing the density of the maximum eigenvalue from a single 
4 In (Smith and Grath, 2007), a similar analysis has been performed 
to test the MIMO (Multiple Input Multiple Output) channel transitions 
probability. 
polarimetrie acquisition (complex Wishart distribution). In (Mar 
tinez et al., 2005), the same analyze has been performed by nu 
merical integration. Here, the closed form expression of the pdf 
is given using the teorem 2 in (McKay and et. al, 2007). 
Theorem 2 (McKay and et. al, 2007): Let X ~ Af c (Q n xq, £ <8> 
£2), where q < n, and £2 G c qxq and £ G C nXn are Hermitian 
positive-definite matrices with eigenvalues wi < < w q and 
<7i < < cr n , respectively. Then the pdf of the maximum 
eigenvalue A max of the complex Wishart matrix X f X is given 
by 
t (15) 
Z=t+1 / 
where \D(:r) is a n x n matrix with (i, j)th element 
... I (*«*r 
Wx)ki ip ( „, 
Wi-tVj 
w i~t eT j 
i ^ l, t = n — q 
, i — l 
(16) 
and where 
i'ï'(æ)i = I ( a i ) 
' 7 i.i I nitrii - 
(n-i) 
exp( —)V(n, —), i > t 
Wj_ t CTj > A > Wi_t<Tj /’ 
i < t, t = n — q 
(17) 
Figure 1: Distribution of the maximum eigenvalue as a function 
of the number of looks for ¿1 = 3, /2 = 2, /1 = 1. When 
n ^ OO, X'max — ^1 “ 3. 
3 A STATISTICAL TEST WITH APPLICATIONS 
In previous sections, to investigate the temporal behavior of po 
larimetrie data, the joint and the marginal density functions of 
eigenvalues from temporal images were derived in the context of 
target decomposition theorem. In this section, previous results 
are discussed considering potential applications.