The Use of Multidimensional Copulas to Describe
Amplitude Distribution of Polarimetric SAR Data
Grégoire M ERCIER† , Lynda B OUCHEMAKH‡ and Youcef S MARA‡
† GET
/ ENST Bretagne / dpt. ITI
CNRS UMR 2872 TAMCIC / TIME
Technopole Brest-Iroise,
CS 83818, F-29238 Brest Cedex 3, France
‡ Lab.
Traitement d’Images et Rayonnement
Faculté d’Electronique et d’Informatique
Université des Sciences et Technologie
Houari Boumediene (USTHB)
BP32 EL Alia, Bab Ezzouar, 16111, Alger, Algerie.
Abstract— The paper focuses on a flexible model of multidimensional probability density function (pdf) dedicated to describe
amplitude distribution of polarimetric SAR data.
The model is based on the copula theory for characterizing the
dependency between polarimetric channels (HH, VV, HV/VH or
the target vector components). The benefit in using copula theory
is to extend correlation concept to a wider dependence one, which
may not be linear. From this point of view, the model is more
flexible than the classical Wishart distribution. But it may include
it.
The other benefit in using the copula model is to separate
the dependence concept from the shape of the marginal pdfs.
Hence, this multidimensional characterization may be linked to
classical 1D Gamma pdf, or to a more flexible Pearson system of
distributions. In the case of high resolution data, pdf shapes are
becoming of heavy tailed and the Fisher system of distributions
seems to be an interesting alternative for such a model. Any
parametric 1D model may be used.
The paper mainly focuses on the model itself and more
precisely on the technique required to construct such multidimensional dependence function. The difficulties arise for copula
on 3D in which the dependency is not homogeneous between the
components (the link between HH and VV may not be of the
same behavior as the one between HH and HV).
Illustrations are given on classification and despeckling. Classification will be performed by a Stochastic Estimation Maximisation (SEM). Despeckling will be achieved by a Maximum A
Posteriori technique.
I. I NTRODUCTION
Speckle noise has been intensively studied in SAR data
processing. Since Goodman in 1976 [1] many articles have
been published to describe the multiplicative nature of the
amplitude and the additive nature of the phase. In singlechannel SAR systems, speckle noise may be considered to
as a solved problem, even for high-resolution purpose. Unfortunately, extension to multi-channel SAR application (i.e. in
polarimetric or interferometric domains) is not straightforward.
The speckle nature of the polarimetric data has been mainly
described by Lòpez-Martı́nez [2], [3].
The scattering Sinclair matrix [S] is defined for each
resolution cell as
S
SHV
[S] = HH
,
(1)
SVH SVV
where Spq corresponds to a scalar-complex value that characterizes the scattered field from a linear polarization p to a q
one. H and V stand for Horizontal and Vertical polarizations.
[S] characterizes completely the scattering process for a deterministic phenomenon. Unfortunately, it fails to characterize
random process, mainly involved in distributed targets. Then,
[S] can be composed to a target vector k [4] such as :
t
√
k = SHH
(2)
2SHV SVV ,
where t stands for the transpose.
When considering the coherent behavior of SAR system and
the Goodman assumptions, i.e. a Gaussian scattering behavior
assumption, target vector k may be modeled by a multivariate
complex zero-mean Gaussian pdf of covariant matrix C :
pk (k) =
∗
−1
1
e(−k [C] k) ,
π 3 det C
(3)
where k∗ stands for the transposed complex conjugate of
k. The components of the covariance matrix [C] are more
tractable than those of [S] for distributed targets since [C] is
involved in the correlation structure of the data.
On homogeneous area, [C] can be estimated locally through
a spatial averaging (for instance by using a neighborhood of
N × N samples around the current pixel of position (i, j)) :
[Ĉ] =
N X
N
X
1
k∗ km .
N × N n=1 m=1 n
(4)
[Ĉ] has been found to be distributed to
as a Wishart pdf [5],
[6], i.e. the marginal pdfs E |Spq |2 that follow Gamma
law and joint pdfs linked to
apcovariance matrix model of
dependence (i.e. E Spq Sp∗0 q0 / E (|Spq |2 ) E (|Sp0 q0 |2 ) stands
of the correlation between components pq and p0 q 0 ).
Unfortunately, for many reasons, such as a local lack of
stationarity or limited number of elementary scatters, this
model may be view as too limited. In single-channel speckle
models, a K-distribution may be preferred to the original
Gamma pdf (when the elementary scatters are not homogeneous anymore but distributed according to a Gamma law
themself), distributions coming from the Pearson system of
distributions or the Fisher system of distributions may be more
appropriated in high-resolution systems [7], [8], [9]. In multichannel speckle models, the use of a covariance matrix to
describe components dependencies may also be too limited. In
this paper, the use of generalized dependency is investigated
to model the behavior of the target vector.
II. T HEORETICAL DEVELOPMENT
The Copula theory [10] is used to model the dependence
between the components of the target vector k and then to
replace the use of the covariance matrix [C].
A. 3D Copula
Copula model states that any joint cumulative distribution
function (cdf) H(x, y, z) of any set (X, Y, Z) of continuous
random variables (RV) may be written in the form:
H(x, y, z) = C FX (x), FY (y), FZ (z)
where FX (·) (resp. FY (·) and FZ (·)) is the marginal cdf of
X (resp. Y and Z) that can be defined by any 1D parametric
distribution function. C(·, ·, ·) is a unique 3D cdf on [0, 1]3
with uniform marginals. Some articles have been published
on the use of copula for remote sensing application (mainly
for change detection [11], [12] and random Markov field
purposes [13], [14]). All of them concern 2D copulas only.
In fact, most of the parametric copulas have been defined on
2D. The Archimedian copulas are a flexible family of copulas
since they are defined by a 1D continuous decreasing convex
function ϕ(·) from [0, 1] to [0, ∞[ such that ϕ(1) = 0,
C(u1 , u2 , u3 ) = ϕ−1 ϕ(u1 ) + ϕ(u2 ) + ϕ(u3 ) .
Nevertheless, it defines a symmetrical copula only which
does not fit polarimetric data, since, for instance, dependence
between SHH and SVV is not of the same kind as the one
between SHH and SHV .
A combination of 2D-copulas does not induced a multidimensional copula systematically [15]. When using Archimedian copulas, it is possible to draw a multidimensional copula
by limiting the nature of the dependence structure with 2Dcopulas [16]. Hence, the 3D-copula model may be written as:
C(u1 , u2 , u3 ) = C1 (u3 , C2 (u1 , u2 ))
= ϕ−1
ϕ1 (u3 ) + ϕ1 ϕ−1
. (5)
1
2 (ϕ2 (u1 ) + ϕ2 (u2 ))
In general, n variables may be coupled in n(n−1)
ways (the
2
number of bi-variate margins). The previous model is just one
solution from all of those. The combination of the ϕi has to
follow some theoretical restrictions that induce the initial data
x, y, z to be linked to the components u1 , u2 , u3 by degree of
dependence among them.
This model is applied at each resolution cell to model the
statistical behavior of the target vector k with respect to a
sliding estimation window of size w × w. At each position
of the sliding window, parameters of the 3D-copula have
to be estimated as well as the parameters of the marginal
pdfs. The benefit in using the copula theory is to split the
dependence concept and the marginal pdf. So that marginal
pdf can be chosen to be coming from any parametric model
such as a gamma law, the Pearson or even the Fisher system
of distribution. In fact, the initial pdf of k stated at eq. (3) may
be rewritten by using a Gaussian copula and Gamma marginal
pdfs.
B. Goodness of fit
The K-plot [17] is a rank based graphical tool for visualizing
dependence between two RVs. The technique is based on the
plot of the pairs Wn:N , H(n) for n ∈ {1, 2, . . . , N }.
• H(n) is defined as H(1) 6 H(2) 6 · · · 6 H(N ) , i.e. the
order statistics of the quantities Hn defined as
Hn =
1
card {m 6= n : xm 6 xn and ym 6 yn } .
N −1
• Wn:N corresponds to the expected value of the n-th statistics from a random sample of size N of the RV W =
H(X, Y ) (or W = C(U, V ) which is the same) under the
null hypothesis of independence between the two components. It is given by:
N −1
Wn:N = N
n−1
Z 1 n
on−1 n
oN −n
w K0 (w)
1 − K0 (w)
dK0 (w), (6)
0
where K0 (w) = P(U V 6 w) = w − w log w.
It is possible to compare K0 (w) to the parametric distribution
Kθ (w) which depends on the dependence between the two RV,
and so, the parameter(s) θ of the copula used for modelling.
As for eq. (6), the n-th statistics of W = Cθ (U, V ) may be
computed through eq. (6) in substituting K0 (w) to Kθ (w),
where Kθ (w) = P(Cθ (U, V ) 6 w). From [18], it can be
stated that
Kθ (w) = w −
ϕθ (w)
ϕ0θ (w)
w ∈]0, 1[
for an Archimedian copula of generator ϕθ (·). Many copulas
have been investigated for such a goodness-of-fit test. The Ali
Mikhail Haq copula has been found to fit the best dependence
between components in the target vector. As shown on fig. 1,
two homogeneous areas have been used to perform parametric
estimations. The first one includes fields, the latter a part of
the urban area. For both of them, the same copula gives the
best results. It seems that this copula fit well the dependence
of a pair of SAR images in many cases [11]. Ali Mikhail Haq
copula is generated by: ϕθ (t) = − ln 1−θ(1−t)
, −1 6 θ < 1
t
and takes the following expression, in 2D:
u1 u2
C(u1 , u2 ) =
.
(7)
1 − θ(1 − u1 )(1 − u2 )
Its parameter θ can be estimated by solving the following
expression that uses the Kendall’s τ :
3
8
θ
8 θ
+
τ≈
+ ···
(8)
9 4 − θ 15 4 − θ
Dependence between HH and VV in a
landscape area
Dependence between HH and VV in
an urban area
(a) copula-based SEM segmentation
(b) Wishart H/α classification
Fig. 3.
SEM (with no post processing) and the Wishart-based H/α
segmentation.
Dependence between HH and HV in a
landscape area
Dependence between HH and HV in
an urban area
Dependence between VV and HV in a
landscape area
Dependence between VV and HV in
an urban area
distributions as well as the parameters of the 3D copula. Fig. 3(a) shows such the segmentation yielded by a 3D pdf model
built with marginals issued from gaussian distributions and
the Ali Mikhail Haq copula. It is worth noting that copulabased approach seems to be more selective from landslide
regions (i.e. distributed targets) instead of the Wishart-based
H/α (fig. 3-(b), performed by PolSARpro v3.0).
B. Copula-MAP
Fig. 1. Graphical goodness-of-fit. The Ali Mikhail Haq copula has been
found to fit the best dependence between components in the target vector.
This model is valid on landscape area as well as in urban area. The diagonal
lines on the graphs characterizes the independence between the two RVs.
III. A PPLICATION TO P OLARIMETRIC SAR DATA
This 3D pdf model has been applied on a set of ESAR data
(channels HH, VV, HV), shown on fig. 2.
A. Unsupervised segmentation
The Stochastic Estimation Maximization (SEM) segmentation procedure is a stochastic version of the classical EM
algorithm where estimation of parameters from given samples
(E stage) and estimation of the probability of each class
(M stage) are iteratively processed. In this application, the
parameters to be estimated come from the ones of the marginal
In single channel application, one can consider that the
conditional distribution f (I|R) of a SAR image follows a Generalized Gamma pdf (when considering amplitude distribution
f (A|R) becomes a Rayleigh Nakagami). Under the hypothesis
which assumes that the reflectivity R follows a Gamma pdf,
one can derive easily the Gamma-MAP filter. This is achieved
by the maximization of the likelihood of f (R|I), by using the
log-derivation annealing:
R̂
/
∂
∂
log f (I|R) +
log g(R) = 0.
∂R
∂R
(9)
When using the target vector in 3D, multidimensional conditional pdf may be modelled with a copula:
f (k|r) = f1 (k1 |r1 )f2 (k2 |r2 )f3 (k3 |r3 )
c F1 (k1 |r1 ), F2 (k2 |r2 ), F3 (k3 |r3 ) .
It is assumed in this paper that the components of reflectivity
follow the same dependence rules. Then
g(r) = g1 (r1 )g2 (r2 )g3 (r3 ) c G1 (r1 ), G2 (r2 ), G3 (r3 ) .
Fig. 2.
Initial Original ESAR image. red=s11, green=s22, blue=s12.
Note that the density of the copula is considered here and not
the copula itself which is a cdf. When considering components
independently, the density of the copula c(·, ·, ·) is constant and
equals 1. If gi (ri ) is considered to follow a Gamma pdf, the
resolution of eq. (9) yields the classical Gamma-MAP filter on
the three components r1 , r2 and r3 . If gi (ri ) is considered to
follow a inverse Gamma law, it comes the Fisher-MAP [19].
When using the copula built on eq. (5), the multidimensional
MAP-filter becomes:
t
r̂ = (r̂1 , r̂2 , r̂3 )
/ ∀i ∈ {1, 2, 3}
(10)
∂
∂
log fi (ki |ri ) +
log gi (ri )
∂ri
∂ri
∂
log c F1 (k1 |r1 ), F2 (k2 |r2 ), F3 (k3 |r3 )
+
∂ri
∂
+
log c G1 (r1 ), G2 (r2 ), G3 (r3 ) = 0.
∂ri
Unfortunately, the log-derivations of the density of the copula
is not easy to write with the copula defined on eq. (5).
Nevertheless, their expressions remain easy to program.
The differences between eq. (9) and (10) arise in the
∂
∂ri log c(·, ·, ·) terms. Those terms induce a multicomponent
constraint to the initial scalar MAP filter. As soon as the
annealing of eq. (10) may be complex, the strategy that has
been adopted is to perform a first guess on ri by using eq. (9)
for i ∈ {1, 2, 3} and then uses those first guesses for the value
of the ∂r∂ i log c(·, ·, ·) terms. Let Ci be defined as
∂
log c F1 (k1 |r1 ), F2 (k2 |r2 ), F3 (k3 |r3 )
∂ri
∂
log c G1 (r1 ), G2 (r2 ), G3 (r3 ) ,
+
∂ri
for which, the parameters of the marginal cdfs are estimated
by using classical Generalized Gamma law on Fi (ki |ri ) of
parameter (Li ) and Gamma or Inverse Gamma hypothesis on
Gi (ri ), of parameters (µi and Mi ). Then, the Copula-GammaMAP is defined by:
ri = µi (Mi − Li − 1)
q
+ µ2i (Mi − Li − 1)2 + 4µi Li ki (Mi + µi Ci )
Ci =
/2(Mi + µi Ci ).
(11)
The Copula-Fisher-MAP is defined by:
ri = − (Li + Mi + 1)
p
+ (Li + Mi + 1)2 + 4Ci (Li ki + Mi µi )
/2Ci .
(12)
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Fig. 4 shows the results of a Fisher-MAP and a Copula-FisherMAP.
IV. C ONCLUSION
Copula framework seems to be an valuable alternative
for modelling multidimensional pdfs with generalized dependence. Its use in polarimetric domain induces new perspective
for distributed targets. At present time, physical interpretation
is required to deeper develop processing of polarimetric data.
R EFERENCES
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Amer., vol. 66, no. 11, pp. 1145–1149, Nov. 1976.
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model,” IEEE Trans. Geosci. Remote Sensing, vol. 41, no. 10, pp. 2232–
2242, Oct. 2003.
Fisher MAP
Copula MAP
Fig. 4. Results of a fisher MAP and a copula-fisher-MAP with sliding window
of size 9 × 9.