IEEE Geoscience and Remote Sensing Magazine - September 2013 - 9
II. SIGNAL AND NOISE MODELING
Under a statistical signal processing perspective, despeckling filters aim at estimating the noise-free radar reflectivity from the observed noisy SAR image [20]. In order to
describe the estimation methods that have been developed
for the despeckling problem, we need firstly to introduce
models for speckle, SAR system and reflectivity.
A. Speckle modelS
SAR is an active acquisition instrument that produces a
radiation and captures the signals backscattered from a
small area of the imaged scene (resolution cell). The received
signal, as output from the in-phase and quadrature channels, is complex. If we assume that the resolution cell contains several scatterers and that no one yields a reflected
signal much stronger than the others (distributed target),
then the received signal can be viewed as the incoherent
sum of several backscattered waves, i.e., Ae jz = / i A i e jz i, as
shown in Fig. 1. The amplitudes A i and phases z i are the
result of several factors, including propagation attenuation,
scattering of the illuminated targets, antenna directivity.
Each individual component, however, can not be resolved
within a resolution cell. A first approach to modeling the
received signal is solving the Maxwell's equations according
to the propagation geometry and scattering medium [24],
[25]. By using this approach, the way each propagation path
interferes gives us basic information about the observed
scene. On the other hand, if we consider that the phases
of each path are highly different and that they may sum
in a constructive or destructive way, then the amplitude of
the received signal varies randomly. So, even if the underlying reflectivity field is uniform, it appears as affected
by a "granular" noise after the imaging system. For visual
inspection and for specific applications that involve visual
information retrieval, such as mapping and segmentation,
the highly varying nature of the signal may be considered as
a disturbance and is commonly denoted as "speckle".
The phases z i are highly varying (since the wavelength
is much shorter than the resolution cell size and scatterers
distances) and may be considered as uniformly distributed
in (-r, r) as well as independent of A i . If the number of
scatterers is sufficiently high, the central limit theorem
applies [26] and the resulting signal Ae jz = z 1 + jz 2 can be
seen as a complex signal whose real and imaginary parts
(in-phase and quadrature components) are independent
and identically distributed zero-mean Gaussian variables
with variance v/2. When this applies speckle is termed as
fully developed [27]. The joint probability density function
(pdf) is given by
2
2
z1 + z2
1
p z1, z2 (z 1, z 2) = rv e - v ,
(1)
whereas the amplitude A is distributed as a Rayleigh pdf,
that is
2
2A A
p A (A) = v e - v
September 2013
(2)
ieee Geoscience and remote sensing magazine
and the power or intensity I = A 2 is distributed according to
an exponential pdf, that is
1 I
p I (I) = v e - v
(3)
so that the mean of the intensity is equal to v. It can be
shown [4], [28] that the intensity measurement carries
information about the average backscattering coefficient
(for distributed targets) related to the resolution cell.
Hence, for specific applications, the parameter v is the
actual information we would like to extract from a single
channel SAR system. This can be considered as the radar
cross section (RCS) of the observed resolution cell. The
received signal pdf can be reformulated into
1 I
p I v (I v) = v e - v
(4)
I = vu ,
(5)
or
where u is exponentially distributed, that is,
p u (u) = e -u .
(6)
Eq. (5) is termed the multiplicative model of speckle.
If only one image (realization of the stochastic process)
is available, the best estimate of the scene average reflectivity is just the pixel-by-pixel intensity. This will be a quite
noisy estimate because of the previously described constructive/destructive combination effects. From (3), it follows that the variance of the intensity in each pixel is v 2,
so that brighter pixels will be affected by stronger disturbances than darker ones. A way to improve the estimation
of v is to average L independent intensity values related to
the same position. This processing, named "multilooking",
maintains the mean intensity v but reduces the estimator
variance to v 2 /L. Independent "looks" of a target resolution cell can be obtained either by appropriate processing
in the Doppler domain (splitting the Doppler bandwidth
within the imaging system that compensates the quadratic
phase variation created by the platform movement) or by
averaging L spatial observations. In both cases, the cost to
be paid for estimation accuracy improvement is spatial resolution loss by a factor L. If the hypothesis of independent
intensity measurements holds (in the case of correlated
data the assumption fails), the L-look averaged intensity
I L is C-distributed, that is
p IL v (I L v) =
1 ` L jL L - 1 - LIvL
I
e
C (L) v L
(7)
whereas the relative amplitude image A L = I L has a square
root C distribution [4]. For visual inspection and for automatic interpretation tasks, the use of amplitude images is
preferable, thanks to their reduced dynamic range with
respect to intensity images, which is accompanied by an
increment in SNR.
9
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