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.