IEEE Geoscience and Remote Sensing Magazine - September 2013 - 21

6

WfMAP

4
2
0

-2
-4
-6
-6

-4

-2

0
Wg

2

4

6

2

4

6

(a)
6
4

WfMAP

2
0

-2
-4
-6
-6

-4

-2

0
Wg
(b)

tf
Figure 9. Mapping of the W

MAP

estimates vs the observed W g: in
(a) v W f = 2, v Wv = 1, o Wv = 2 and o W f varies from 0.4 to 2 with
step 0.2; in (b) o Wv = 1.2 (the other parameters are unchanged).
where C is the Gamma function, v is the standard deviation of the distribution, v is a shape factor, and h (o, v) is
given by
1 C (3/o) F
h (o, v) = v <
.
C (1/o)
1/2

(30)

The GG distribution is reasonably simple, since the use of
only two parameters allows different levels of "peakedness"
to be achieved. As particular cases, the GG pdf includes
both the Laplacian and the Gaussian pdfs, for o = 1 and
o = 2, respectively. A plot of GG pdf curves for different
values of o is shown in Fig. 8.
Substituting (29) into (28) yields
hW f oW f
t MAP
W
= arg max ;ln 2C (1/o ) - ^h W f W f hoW
f
Wf
Wf
h Wv o Wv
ln 2C (1/o ) - ^h Wv W g - W f ho W E .
Wv
f

v

(31)

In [82], a method for the estimation of the parameters
relative to the GG model, i.e., the standard deviation v
and the shape factor v of the distributions relative to
W f and W v, is given. The estimation of the parameters
is based on the computation of some moments of the
observable variables g and W g . In the implementation
September 2013

ieee Geoscience and remote sensing magazine

of the filters, these moments are substituted by spatial
averages. The solution of equation (31) is not known in a
closed analytical form and a numerical optimized solution has been proposed in [82].
t MAP
In Fig. 9, a set of curves plotting W
vs W g is given
f
for particular values of the parameters of the GG model: in
Fig. 9-(a), the curves refer to v W f = 2, v Wv = 1, o Wv = 2 and
to o W f varying from 0.4 to 2 with step 0.2; in Fig. 9-(b), the
parameter o Wv has been changed to 1.2 (the other parameters were not modified). Such curves define a remapping
of the observed coefficients onto noise-free ones same as it
is done by hard and soft-thresholding schemes commonly
used for denoising signals affected by additive signalindependent noise [67], [132]. It is important, however,
to point out that for despeckling the wavelet coefficients
are modified according to the multiplicative model of
speckle and thus adaptively vary according to the locally
estimated parameters.
MAP-LG filter-In [85], the empirical distribution
of the shape factor of noise-free reflectivity coefficients
has been investigated and an interesting behavior was
noticed. For the lowest levels of decomposition, the shape
factor is usually very close to one, whereas it tends to shift
towards two in highest ones. The shape factor of signal-
dependent noise coefficients, instead, are mostly concentrated around the value two. These facts suggest directly
introducing a combination of Laplacian and Gaussian
pdfs into the modeling: this yields some computational
advantages with respect to using the more general GG
pdf. In fact, by assuming that the wavelet coefficients W v
and W f follow a zero-mean Gaussian and zero-mean
Laplacian distribution, respectively, yields the following
closed form estimator [133]:
t MAP
W
= arg max p W f W g (W f W g)
f
Wf
Z W g - t,
if W g > t
]
= [ W g + t,
if W g < - t
]
otherwise
0
\

(32)

where t = 2 v 2Wv /v W f . Thus, the estimator is equivalent to
a soft-thresholding algorithm with a locally adaptive threshold. Eq. (32) has been originally devised in [134] and used
for processing ultrasound images with decimated wavelets.
C. Adjustments for sAr imAge heterogeneity
In several despeckling methods, different filtering strategies are used according to the texture content of the scene.
In [51], [54], the coefficient of variation is used to discriminate among homogeneous, textured and highly heterogeneous (or point target) areas. Pixel belonging to the
first two classes are filtered by using simple averaging and
C-MAP, or another local-statistics filter, whereas no filtering is attempted on point targets. A strongly scattering
target, however, is concentrated in space, but after wavelet analysis its response will be somewhat spread because
of the finite support of the wavelet function. Thus, also
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