Aerospace and Electronic Systems - June 2019 - 31

Addabbo et al.

Figure 3.
Results from [9] on L-Band AirSAR data with the H=a classification (left) and the symmetric H=a classification (right) methods.

are used to synthesize classification architectures.
Finally, the illustrative examples, built up using both
simulated and measured RADARSAT-2 SAR data,
show the effectiveness and usefulness of the proposed
algorithms.
The remainder of this paper is organized as follows.
Section "PARAMETERS DEFINITION AND DATACUBE CONSTRUCTION" provides some preliminary
definitions, while Section "PROBLEM FORMULATION
AND CLASSIFICATION STRATEGIES" is devoted to
problem formulation, covariance estimation under each
hypothesis, and the design of the MOS rules. In Section
"ILLUSTRATIVE EXAMPLES AND DISC," the behavior of the proposed algorithms is assessed adopting the
probability of correct classification as the performance
metric. Finally, Section "CONCLUSION" draws conclusions and traces future research works.

NOTATION
The adopted notation uses boldface for vectors a (lower
case) and matrices A (upper case). CNÂM is the set of
complex matrices of size N Â M (or vectors if M ¼ 1).
A diagonal matrix whose nonzero entries are the components of a vector a is denoted by Diag ðaaÞ. The
transpose and the conjugate transpose are denoted by
ðÁÞT and ðÁÞ{ , respectively. tr fÁg and detðÁÞ are the trace
and the determinant of the square matrix argument,
respectively. I denotes the identity matrix, whose size
is determined from the context, and k Á k denotes the
Euclidean vector norm.

different polarimetric channels (namely HH, HV, and VV).
The N returns associated with the same pixel are organized
in the specific order HH, HV, and VV to form the 3-D vector
x l;m 2 C3Â1 , l ¼ 1; . . . ; L and m ¼ 1; . . . ; M (L and M are
the vertical and horizontal size of the image, respectively).
Therefore, the sensor provides a 3-D data stack X 0 of size
M Â L Â N, which is referred to in the following as datacube (see Figure 4). Starting from the datacube of the illuminated scene, for the generic pixel under test, a rectangular
neighborhood A of size K ¼ W1 Â W2 ! N is extracted.
Finally, let X ¼ ½x 1 ; . . . ; xK 2 C3ÂK be a matrix whose
columns are the vectors of A.

PROBLEM FORMULATION AND CLASSIFICATION
STRATEGIES
The polarimetric returns x 1 ; . . . ; xK are modeled as zeromean circularly symmetric complex Gaussian vectors
whose covariance matrix is s 2k C , k ¼ 1; . . . ; K [12].
Thus, the probability density function (pdf) of x k can be
written as
fðx
xk ; s 2k ; C Þ ¼

1
CÞ
ðps 2k Þ3 detðC

n
h
io
{
À1
:
exp Às À2
k tr C x k x k
(1)

PARAMETERS DEFINITION AND DATACUBE
CONSTRUCTION
A multipolarization SAR sensor, for each pixel of the image
under test, measures N ¼ 3 complex returns for each rangeazimuth resolution cell, which are collected from three
JUNE 2019

Figure 4.
Datacube for polarimetric images.

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Aerospace and Electronic Systems - June 2019

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