Geology Reference
In-Depth Information
This can be obtained from a training set if a supervised
classification scheme is pursued. Equation (10.8) indi-
cates that the a priori probability has less effect on the
classification results as the number of looks increases. In
most applications the a priori probability is unknown and
can therefore be assumed equal for all classes. In this case
it becomes constant (similar to
n
) so it does not affect the
calculations of the relative distances between a sample
coherency matrix ⟨
T
⟩ and each cluster mean. It can then
be removed from the equation, along with
n
. Equation
(10.8) can take the form
data, i.e., not just selected decomposition parameters,
is used. A commonly used matrix for this purpose is the
coherence matrix [equation (7.91)]. Recall that the covari-
ance matrix is related to the coherence matrix [equation
(7.92)]. Using any one of these two matrices should lead
to the same answer. Since most classification approaches
are based on the Bayesian rule [equation 10.1] the param-
eterization of the probability distribution of the complex
coherence matrix should be known. The sample distribu-
tion of the complex multilook coherence or covariance
matrix is determined in
Lee et al
. [1994] to follow the
complex Wishart distribution (formulated by the British
statistician John Wishart in 1928). For
n
look coherence
matrix ⟨
T
⟩ the Wishart distribution of the matrix values
for a given surface is given by the following equation as
reported in
Lee et al
. [1999]:
1
dTVVVT
m
,
ln
Tr
(10.10)
m
m
A classification scheme based on this distance is called
the Wishart classifier [recall that Wishart distribution
was used to derive equation (10.8)]. This classifier is read-
ily available in open‐source or commercial software for
processing polarimetric data. It is also easy to code. A
pixel is assigned to class
ω
m
if
nq
n
n
qn
1
pTnT nTrV TKnq V
T
exp
/
,
(10.5)
where ⟨
T
⟩ is the average of the
n
‐independent single‐look
coherence matrix
dTVdTV
m
,
,
(10.11)
j
j
m
1
n
T
T
ukuk
(10.6)
n
A few studies have used the Wishart classifier to classify
sea ice types [
Lee et al
., 1994;
Scheuchl et al
., 2001;
Brath
et al
., 2013]. Based on Wishart distribution,
Dabboor and
Shokr
[2013] developed a new likelihood ratio, called the
Bayesian likelihood ratio test (BLRT), for the supervised
classification of polarimetric data. The proposed BLRT
test is based on the asymptotic classification error prob-
ability derived by
Chernoff
[1952]. It is complemented
with another new spatial criterion to incorporate the
spatial context information in order to produce more
homogeneous classes. The authors applied the method
to classify four ice types in the area of Franklin Bay,
Canadian Arctic, using fully polarimetric C‐band
Radarsat‐2 data. Results show that the combined use of
the BLRT test and the spatial criterion produces homog-
enous classification results with better accuracy than
using the Wishart classifier.
SAR polarimetric data has proven to be more effective
for sea ice classification than single or dual polarization.
However, the data have been available so far only from
narrow or standard swath modes (up to 50 km wide from
Radarsat‐2 quad‐polarization mode). This is too small a
width for operational applications that usually utilize
images from wide‐swath modes of 300-500 km wide. A
significant advancement in the field of SAR architecture
has been achieved by developing a hybrid polarity system
known as compact polarimetry (CP) [
Raney
, 2007]. The
definition is included in section 7.6.2.3, but it suffices to
reiterate here that it refers to a system that transmits a
k
1
In equation (10.5),
V
=
E
[⟨
T
⟩] is the expected (average)
value of
T
over all pixels, Tr is the trace of a matrix,
q
= 3 for the reciprocal SAR (i.e., under the assumption
S
hv
=
S
vh
), and
q
= 4 for the bistatic SAR. The normaliz-
ing factor
k
is defined as
1
2
qq
1
Knq
,
n
,
nq
1
(10.7)
where
Γ
is the gamma function. The Wishart distribution
equation [equation (10.5)] can be used directly in a
Bayesian maximum‐likelihood approach [equation (10.1)],
but it can also be used to derive a distance measure based
on the Bayesian equation.
Lee et al
. [1994] derived the
following expressions for a distance measure between a
sample coherency matrix ⟨
T
⟩ and a cluster mean of the
m
th
class
V
m
:
dTVnV
,
ln
Tr
V T
1
ln
P m
(10.8)
m
m
m
where
P
(
m
) is the a priori probability of class
m
th
and
V
m
is class mean of ⟨
T
⟩ for all pixels in the
m
class (
ω
m
):
VETT
m
m
(10.9)
