IEEE Geoscience and Remote Sensing Magazine - September 2013 - 11

III. SPACE AND SCALE-SPACE DOMAIN ESTIMATION
From the previous discussion, it emerges that modeling the received SAR signal should take into account
several physical, statistical and engineering aspects of
the overall system. Such a complexity makes the process of extracting average backscatter information from
the observed signal a nontrivial task. From a signal
processing perspective, a first step towards finding efficient solutions is stating the acquisition model in the
simplest form as possible. In [20], several multiplicative
models of speckle are described and classified according to the autocorrelations of the imaged scene and of
the noise term.
In the following of this section, models of the noisy signal in both spatial and transformed domains are reviewed,
Bayesian estimation principles are briefly recalled and the
wavelet transform, in both decimated and undecimated
versions, is introduced as a transformation suitable for
despeckling. Eventually, the modeling of pdfs for Bayesian
estimation in the wavelet domain is discussed and shown
to be crucial for performances.
A. Models of noisy signAl
Perhaps, the most used model in the literature on despeckling is the following:
g = fu ,

(13)

where v = (u - 1) f accounts for speckle disturbance in an
equivalent additive model, in which v, depending on f, is a
signal-dependent noise process.
A second way that allows the multiplicative noise to be
transformed into an additive one is using a homomorphic
transformation [32]. It consists of taking the logarithm of
the observed data, so that we have
log g = log f + log u
g l = f l + ul ,

(15)

where g l , f l and ul denote the logarithm of the quantities in (13). Unlike the case in (14), here the noise
component ul is a signal-independent additive noise.
However, this operation may introduce a bias into the
denoised image, since an unbiased estimation in the logdomain is mapped onto a biased estimation in the spatial domain [33]; in math form, if u exhibits E [u] = 1,
E [ul ] = E [log (u)] ! log (E [u]) = log (1) = 0.
Over the last two decades, approaches to image denoising that perform estimation in a transformed domain have
been proposed. Transforms derived from multiresolution signal analysis [34], [35], such as the discrete wavelet
transform (DWT), are the most popular in this context.
Despeckling in a transform domain is carried out by taking
the direct transform of the observed signal, by estimating
the speckle-free coefficients and by reconstructing the filtered image through the inverse transform applied to the
despeckled coefficients.

where f is a possibly autocorrelated random process
and represents the noise-free reflectivity; u is a possibly autocorrelated stationary random process, indeB. BAyesiAn estiMAtion concepts
pendent of f, and represents the speckle fading term; g
From the previous discussion about the most widely used
is the observed noisy image. All the quantities in (13)
signal models for despeckling, it can be seen that the multimay refer to either intensity or amplitude as well as to
plicative model is often manipulated in order to obtain an
single-look or multilook images, whose pdfs have been
additive one. Fig. 2 summarizes the various versions of the
described previously.
additive models.
The variable u may be assumed as spatially correlated [30].
The block "Estimator" attempts to achieve a speckleRecently, it has been shown [31] that a preprocessing step that
free representation of the signal in a specific domain; for
makes speckle uncorrelated, that is "whitens" the complex
signal, allows despeckling algorithms
designed for uncorrelated speckle to be
successfully applied also when speckle
is (auto)correlated. Therefore, in the
g=f+v
Estimator
fc
following we shall analyze only algo(a)
rithms working under the hypothesis of
exp
g = fu
Estimator
log
uncorrelated speckle.
fc
The nonlinear nature of the rela(b)
tionship between observed and
g=f+v
w -1
w
Estimator
fc
noise-free signals makes the filtering
procedure a nontrivial task. For this
(c)
reason, some manipulations have
exp
log
g = fu
Estimator
w
w -1
fc
been introduced to make the observa(d)
tion model simpler. Several authors
adopt the following model, derived
from (13):
FIgurE 2. Additive models commonly used in despeckling algorithms: (a) signal-dependent
in spatial domain, (b) signal-independent in spatial domain, (c) signal-dependent in transg = f + (u - 1) f = f + v,
(14) form domain, and (d) signal-independent in transform domain.
September 2013

ieee Geoscience and remote sensing magazine

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