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The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part Bl. Beijing 2008
after transforming An.2 into An(I — R 2 ), the joint density may
be given in its final form 2
p (An. A22, R 2 )
, ( PVE^Eri a„a 22 r 2 \
oFi r i-p 2 rcvT-)
etr
'S22 A22 + Sji 1
I-P 2
n pn \I- R 2 | n - p |AiiA 22 | n ~ <?
|S| n fq(n - q)f q (n)T q (q)
Here, P 2 = Si 2 E^ 1 1 E^ 2 1 E 2 i, 0-Ft is the complex hyperge
ometric function of matrix argument, and f q (n) is a complex
gamma function
t q (n) = TT q( ' q 1)/2 PJ T(n — i + 1). (4)
¿=1
It is clear that (3) is valid for 0 < P 2 < I, which means that
both |P| and |I — P 2 1 are positive definitive. When Si 2 = 0,
then An, A 22 and R are independent. As shown in (3) and well-
known from SAR literature, unbiased characterization of tempo
ral data is related to unbiased estimate of coherence (P R)
and speckle free data (Sn <-» An).
2.2 Derivation of the joint density of temporal eigenvalues
Theorem I (James, A.T., 1964): If p(A)(dA) is the pdf of a
Hermitian complex matrix variate A, then the distribution of the
diagonal matrix W of the latent roots of A, A — UWU\ is
r ra(m — 1) m
/ p(UWU')(dU) , / N TT(^ - w j) 2 dwi....dwm-
J U (m) Г m(m)
(6)
Here, it is important to note that after applying the theorem 1
into (5), the matrix P still remain in the joint eigenvalue distribu
tion. However, it is difficult to foresee the behavior of the density
function or to understand how the eigenvalues interact with each
other in the presence of the matrix P. It makes sense to make
the change of variables Li = HE^ 2 and L 2 = QE2 2 1/2 with
Jacobians ISnl* and |E 22 | 9 to make the matrix P 2 diagonal. It
turns out that
P = E“ 1/2 Si 2 S 22 1/2 = H t PQ
H G O(q), Q G O(q)
O : orthonormal group
(7)
where P 2 is a diagonal matrix consisting of square of canonical
correlation coefficients (1 > p 2 > ... > p q > 0). For the de
tailed analysis about canonical correlation coefficients, we refer
to (Muirhead, 1982) 3 .
P =
pi • • • 0
. 0 • • ■ Pq
While statistical aspects concerning Wishart matrices have been
well developed, there seem to be little work on the eigenvalue
statistics of correlated Wishart process over time. Although in
(Smith and Grath, 2007) and (Kuo et al., 2007) the joint density
of the eigenvalues of correlated Wishart has been derived, both
analysis has been performed based on the assumption that co-
variance matrices (Sn) are unitary and the correlation between
random complex variants are the same (P = pi). However, this
is a too restrictive assumption for the polarimetrie case, since in a
general scattering scenario the covariance matrix of polarimetrie
data is no more unitary, and each polarimetrie channel has arbi
trary correlations ^P = Sj‘ 1 1/2 Si 2 S2 2 1/2 y Due to this missing
analysis, the joint distribution of eigenvalues of temporal polari
metrie images is derived in this section. The main aim of the
analysis of the joint eigenvalue distribution is to study the tem
poral behavior of different scattering mechanism. In addition to
characterization of different scattering mechanism, investigation
of the dimension reduction is the second objective of this analy
sis.
To obtain the joint distribution of eigenvalues of correlated Wishart
matrices, firstly the correlation parameter R must be integrated
from (3). Accordingly, applying the theorem 7.2.10 in (Muir
head, 1982) to (3) follows that
P (An, A 22 )
p („ n 2 P 2 Sp An Е^Аи
oFl ("'—I-P 2 I-P 2
etr (—nE^.iA 22 )efr (-nE“ 1
n pn| I _p2|nq| An |n —д|д 22 |
|Ell.2| n |E¡ 22 .l| n f q(n)f q(n)
11.2
n —
Then, making the transformations An = UjWiU|, A 22 =
U 2 W 2 U.2 and integrating (5) with respect to dUi and dU 2 over
the orthogonal group 0(q), the joint distributions can be obtained
via following theorem,
2 The proof of this distribution for the real case can be found in (Muir
head, 1982) and (Lliopoulos, 2006).
Considering (6) and (7) into (5), p(Ai, A 2 , A3, A",A 2 , A3) =
p(Wi, W2) results
П?<Л(М-^) 2 (А{'-лу) 3 }
7 r<J(l-9)fq(n) 2 n?(l-p?) n
ex p -Ей
ni!-
+
(i-p\) lí'G-p?)
x , x „\n-q
im)
Л i 1 L
П?<Л<‘Т 1 - 1 7 1 > 3 < 1
) 2 }
Ï I 0 Fi (n,EÜTÜ^dÜ
U(q)
(8)
where E and T indicate the matrix parameters of hypergeometric
function in (5) and A', A", and 1” for i = 1, ...,q denote the
eigenvalues of An, A 22 , En and S 22 respectively. Here, it can
be noted that U = U\U2 and (dU) = (dUi) over U E 0(q).
Using the relation (James, A.T., 1964)
[ p F q (sÜTÜ')dÜ = p F q (E,r),
J U (n) 4 '
(9)
the integration part in (8) has been solved. Despite the joint dis
tribution has been derived, it is expressed in terms of an infinite
series (hypergeometric functions) that makes the analysis of the
distribution hard. However, hypergeometric functions of matrix
arguments can be expressed in terms of the matrix eigenvalues
using Zonal Polynomials (Muirhead, 1982). For the specific case
of (10), 0 Fi(.,.), the closed form of the hypergeometric fuction
exits, and it is given by (Gross and Richards, 1989)
oFi(n,s,t) =
riwriw A t ,n-M2 \в[ я - п)/2 1„^(гущ
An
тгя(я-1) J-i k
k=1
UUjisi
i<J '
(10)
Related to this expression, (8) can be solved without the need of
zonal polynomials.
3 In (Muirhead, 1982), only real differential forms have been consid
ered. However, the theorems have been extended to the complex case
after some algebra, e.g. (dz) = dTZ {z} AdX {2} where A, 72 and I
indicate exterior products, real and imaginary parts of the variates respec
tively.
