IEEE Geoscience and Remote Sensing Magazine - September 2013 - 20

on Gamma distributed speckle [120], [124], a combination of the previous terms [121], the generalized Kullback-Leibler divergence [123], the L 1 norm on curvelet
coefficients [122]. It is worth noting that all the above
methods have been mainly validated on simulated data.
The literature regarding the application of such methods to actual SAR images is quite scarce [125]-[127], and
there is a general lack of comparisons with Bayesian and
NL despeckling methods.
Despeckling Based on Compressed Sensing-A new
signal representation model has recently become very
popular and has attracted the attention of researchers
working in the field of restoration of images affected by
additive noise as well as in several other areas. In fact,
natural images satisfy a sparse model, that is, they can
be seen as the linear combination of few elements of a
dictionary or atoms. Sparse models are at the basis of compressed sensing [128], which is the representation of signals with a number of samples at a sub-Nyquist rate. In
mathematical terms, the observed image is modeled as
y = Ax + w, where A is the dictionary, x is a sparse vector,
such that ;x; 0 # K, with K % M, with M the dimension
of x, and w is a noise term that does not satisfy a sparse
model. In this context, denoising translates into finding the sparsest vectors with the constraint ;y - Ax; 22 < e,
where e accounts for the noise variance. The problem
is NP-hard, but it can be relaxed into a convex optimization one by substituting the pseudo-norm ; $ ; 0 with
; $ ; 1 . Recently, some despeckling methods based on the
compressed sensing paradigm and sparse representations
have appeared [129]-[131].
V. MULTIRESOLUTION BAYESIAN FILTERING
In this section, we review some methods recently proposed for despeckling in the undecimated wavelet domain
that use a multiresolution analysis. The methods refer to

the additive model in (26), that is, they do not exploit the
homomorphic transform, which may introduce bias in the
estimation of the despeckled image.
Fig. 7 outlines the flowchart of Bayesian despeckling
in UDWT domain. As it appears, the majority of processing is carried out in the transform domain. Statistics in the
transform domain are directly calculated from the spatial
statistics of the image by exploiting the equivalent filters
(4), as firstly proposed by Foucher et al. [77].
A. LMMSE fiLtEr
In the case of zero-mean Gaussian pdf modeling for the
quantities W f and W v, the MMSE and MAP Bayesian estimators are identical. The expression of the filter has a simple
and closed analytical form that depends only on the space
varying variance of the wavelet coefficients [78], that is

t LMMSE
= Wg $
W
f

2
vW
f
-1 -1
.
2 = W g $ (1 + SNR )
v + vW
v
2
Wf

Thus, LMMSE estimation corresponds to a shrinkage of the
noisy coefficient by a factor inversely related to its SNR.
Unfortunately, the wavelet coefficients of noise-free reflectivity do not respect the Gaussian assumption, especially
in the lowest levels of the wavelet decomposition, so that
its performance are inferior to more complex Bayesian estimators. In Fig. 10-(a) a single-look COSMO-SkyMed image
is shown. In Fig. 10-(b) the despeckled image obtained by
applying the LMMSE estimator is presented.
B. MAP fiLtErS
In this section, we present two different filters that use the
MAP estimation criterion but different models for the pdfs
of the wavelet coefficients relative to the original reflectivity and to the additive signal-dependent noise.
Equation (21) can be rewritten as
t MAP
W
= arg max [ln p WV
f
Wf

3
2.5

o=0.5
o=1
o=2
o=3

pX (x)

2
1.5
1
0.5
0
-3

-2

-1

0
x

1

2

3

FIGURE 8. Zero-mean GG pdfs obtained with unity variance and

different os.
20

(27)

WF

(W g - W f W f ) + ln p WF (W f )].
(28)

Since the signal and noise processes are nonstationary,
space varying pdfs must be considered. The pdfs that are
considered here can be seen as a trade-off between simplicity (few parameters to be estimated from the observed data)
and modeling capability.
MAP-GG filter-In [82], the MAP criterion is combined with a generalized Gaussian (GG) distribution
for the wavelet coefficients. Since the birth of the wavelet recursive algorithm by Mallat [35], a GG pdf has been
used to model image wavelet coefficients and several other
authors use the GG distribution for many image processing tasks involving wavelets. A zero-mean GG pdf depends
only on two parameters and is characterized by being symmetric around the mean. Its expression is given by
o $ h (o, v) - [h (o, v) $| x |] o
Fe
,
2 $ C (1/o)

(29)

ieee Geoscience and remote sensing magazine

September 2013

p X (x) = <



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